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QUADRATIC EQUATIONS - Q&A

EXERCISE 4.1

1. Check whether the following are quadratic equations:

(i) (x + 1)² = 2(x - 3)

Answer:
LHS = (x + 1)² = x² + 2x + 1
RHS = 2(x - 3) = 2x - 6
Equating LHS and RHS:
x² + 2x + 1 = 2x - 6
x² + 2x - 2x + 1 + 6 = 0
x² + 7 = 0
Since this is of the form ax² + bx + c = 0 (where a ≠ 0), it is a quadratic equation.

(ii) x² - 2x = (-2)(3 - x)

Answer:
LHS = x² - 2x
RHS = (-2)(3 - x) = -6 + 2x
Equating LHS and RHS:
x² - 2x = -6 + 2x
x² - 2x - 2x + 6 = 0
x² - 4x + 6 = 0
Since this is of the form ax² + bx + c = 0, it is a quadratic equation.

(iii) (x - 2)(x + 1) = (x - 1)(x + 3)

Answer:
LHS = (x - 2)(x + 1) = x² + x - 2x - 2 = x² - x - 2
RHS = (x - 1)(x + 3) = x² + 3x - x - 3 = x² + 2x - 3
Equating LHS and RHS:
x² - x - 2 = x² + 2x - 3
x² - x² - x - 2x - 2 + 3 = 0
-3x + 1 = 0
This is a linear equation, not a quadratic equation (degree is 1).
No, it is not a quadratic equation.

(iv) (x - 3)(2x + 1) = x(x + 5)

Answer:
LHS = (x - 3)(2x + 1) = 2x² + x - 6x - 3 = 2x² - 5x - 3
RHS = x(x + 5) = x² + 5x
Equating LHS and RHS:
2x² - 5x - 3 = x² + 5x
2x² - x² - 5x - 5x - 3 = 0
x² - 10x - 3 = 0
Since this is of the form ax² + bx + c = 0, it is a quadratic equation.

(v) (2x - 1)(x - 3) = (x + 5)(x - 1)

Answer:
LHS = (2x - 1)(x - 3) = 2x² - 6x - x + 3 = 2x² - 7x + 3
RHS = (x + 5)(x - 1) = x² - x + 5x - 5 = x² + 4x - 5
Equating LHS and RHS:
2x² - 7x + 3 = x² + 4x - 5
2x² - x² - 7x - 4x + 3 + 5 = 0
x² - 11x + 8 = 0
Since this is of the form ax² + bx + c = 0, it is a quadratic equation.

(vi) x² + 3x + 1 = (x - 2)²

Answer:
LHS = x² + 3x + 1
RHS = (x - 2)² = x² - 4x + 4
Equating LHS and RHS:
x² + 3x + 1 = x² - 4x + 4
x² - x² + 3x + 4x + 1 - 4 = 0
7x - 3 = 0
This is a linear equation (degree 1).
No, it is not a quadratic equation.

(vii) (x + 2)³ = 2x(x² - 1)

Answer:
Using identity (a + b)³ = a³ + b³ + 3a²b + 3ab²
LHS = (x + 2)³ = x³ + 8 + 6x² + 12x
RHS = 2x(x² - 1) = 2x³ - 2x
Equating LHS and RHS:
x³ + 6x² + 12x + 8 = 2x³ - 2x
x³ - 2x³ + 6x² + 12x + 2x + 8 = 0
-x³ + 6x² + 14x + 8 = 0
The highest degree is 3 (cubic equation).
No, it is not a quadratic equation.

(viii) x³ - 4x² - x + 1 = (x - 2)³

Answer:
Using identity (a - b)³ = a³ - b³ - 3a²b + 3ab²
RHS = (x - 2)³ = x³ - 8 - 6x² + 12x
LHS = x³ - 4x² - x + 1
Equating LHS and RHS:
x³ - 4x² - x + 1 = x³ - 6x² + 12x - 8
Canceling x³ from both sides:
-4x² + 6x² - x - 12x + 1 + 8 = 0
2x² - 13x + 9 = 0
Since this is of the form ax² + bx + c = 0, it is a quadratic equation.

2. Represent the following situations in the form of quadratic equations:

(i) The area of a rectangular plot is 528 m². The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.

Answer:
Let the breadth of the rectangular plot be x metres.
According to the problem, the length is one more than twice its breadth.
Length = (2x + 1) metres.
Area of rectangle = Length × Breadth
528 = (2x + 1) × x
528 = 2x² + x
2x² + x - 528 = 0
Required Quadratic Equation: 2x² + x - 528 = 0

(ii) The product of two consecutive positive integers is 306. We need to find the integers.

Answer:
Let the first positive integer be x.
The consecutive positive integer will be (x + 1).
Product = x(x + 1)
Given product is 306.
x(x + 1) = 306
x² + x = 306
x² + x - 306 = 0
Required Quadratic Equation: x² + x - 306 = 0

(iii) Rohan's mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan's present age.

Answer:
Let Rohan's present age be x years.
Rohan's mother's age = (x + 26) years.
After 3 years:
Rohan's age = x + 3
Mother's age = (x + 26) + 3 = x + 29
Product of their ages = 360
(x + 3)(x + 29) = 360
x² + 29x + 3x + 87 = 360
x² + 32x + 87 - 360 = 0
x² + 32x - 273 = 0
Required Quadratic Equation: x² + 32x - 273 = 0

(iv) A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.

Answer:
Let the speed of the train be x km/h.
Distance = 480 km.
Time taken = Distance / Speed = 480/x hours.
New speed = (x - 8) km/h.
New time taken = 480/(x - 8) hours.
According to the problem, the new time is 3 hours more than the original time.
480/(x - 8) - 480/x = 3
480 [ 1/(x - 8) - 1/x ] = 3
480 [ (x - (x - 8)) / (x(x - 8)) ] = 3
480 [ 8 / (x² - 8x) ] = 3
(480 × 8) / (x² - 8x) = 3
3840 = 3(x² - 8x)
Dividing by 3:
1280 = x² - 8x
x² - 8x - 1280 = 0
Required Quadratic Equation: x² - 8x - 1280 = 0

EXERCISE 4.2

1. Find the roots of the following quadratic equations by factorisation:

(i) x² - 3x - 10 = 0

Answer:
We need to find two numbers whose sum is -3 and product is -10. These numbers are -5 and 2.
x² - 5x + 2x - 10 = 0
x(x - 5) + 2(x - 5) = 0
(x - 5)(x + 2) = 0
x - 5 = 0 ⇒ x = 5
x + 2 = 0 ⇒ x = -2
Roots are 5 and -2.

(ii) 2x² + x - 6 = 0

Answer:
We need two numbers whose sum is 1 and product is 2 × (-6) = -12. These numbers are 4 and -3.
2x² + 4x - 3x - 6 = 0
2x(x + 2) - 3(x + 2) = 0
(2x - 3)(x + 2) = 0
2x - 3 = 0 ⇒ x = 3/2
x + 2 = 0 ⇒ x = -2
Roots are 3/2 and -2.

(iii) √2x² + 7x + 5√2 = 0

Answer:
We need two numbers whose sum is 7 and product is √2 × 5√2 = 10. These numbers are 5 and 2.
√2x² + 5x + 2x + 5√2 = 0
x(√2x + 5) + √2(√2x + 5) = 0 [Since 2 = √2 × √2]
(√2x + 5)(x + √2) = 0
√2x + 5 = 0 ⇒ x = -5/√2
x + √2 = 0 ⇒ x = -√2
Roots are -5/√2 and -√2.

(iv) 2x² - x + 1/8 = 0

Answer:
Multiplying the entire equation by 8 to remove the fraction:
16x² - 8x + 1 = 0
We need two numbers whose sum is -8 and product is 16. These numbers are -4 and -4.
16x² - 4x - 4x + 1 = 0
4x(4x - 1) - 1(4x - 1) = 0
(4x - 1)(4x - 1) = 0
4x - 1 = 0 ⇒ x = 1/4
Roots are 1/4 and 1/4.

(v) 100x² - 20x + 1 = 0

Answer:
We need two numbers whose sum is -20 and product is 100. These numbers are -10 and -10.
100x² - 10x - 10x + 1 = 0
10x(10x - 1) - 1(10x - 1) = 0
(10x - 1)(10x - 1) = 0
10x - 1 = 0 ⇒ x = 1/10
Roots are 1/10 and 1/10.

2. Solve the problems given in Example 1.

Answer:
Problem (i): John and Jivanti have 45 marbles... (from Example 1 text)
Equation derived in Example 1: x² - 45x + 324 = 0
We need two numbers whose sum is -45 and product is 324. These numbers are -36 and -9.
x² - 36x - 9x + 324 = 0
x(x - 36) - 9(x - 36) = 0
(x - 9)(x - 36) = 0
x = 9 or x = 36.
If John has 9, Jivanti has 36. If John has 36, Jivanti has 9.
They started with 9 and 36 marbles.

Problem (ii): Cottage industry toys... (from Example 1 text)
Equation derived in Example 1: x² - 55x + 750 = 0
We need two numbers whose sum is -55 and product is 750. These numbers are -30 and -25.
x² - 30x - 25x + 750 = 0
x(x - 30) - 25(x - 30) = 0
(x - 25)(x - 30) = 0
x = 25 or x = 30.
The number of toys produced on that day was either 25 or 30.

3. Find two numbers whose sum is 27 and product is 182.

Answer:
Let the first number be x. Then the second number is (27 - x).
Product = x(27 - x) = 182
27x - x² = 182
x² - 27x + 182 = 0
We need two numbers whose sum is -27 and product is 182. Numbers are -13 and -14.
x² - 13x - 14x + 182 = 0
x(x - 13) - 14(x - 13) = 0
(x - 14)(x - 13) = 0
x = 14 or x = 13.
The two numbers are 13 and 14.

4. Find two consecutive positive integers, sum of whose squares is 365.

Answer:
Let the integers be x and x + 1.
Sum of squares = x² + (x + 1)² = 365
x² + x² + 2x + 1 = 365
2x² + 2x + 1 - 365 = 0
2x² + 2x - 364 = 0
Divide by 2:
x² + x - 182 = 0
Factors of -182 summing to 1 are 14 and -13.
x² + 14x - 13x - 182 = 0
x(x + 14) - 13(x + 14) = 0
(x - 13)(x + 14) = 0
x = 13 or x = -14.
Since integers are positive, x = 13.
Consecutive integer = x + 1 = 14.
The integers are 13 and 14.

5. The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.

Answer:
Let the base be x cm.
Altitude = (x - 7) cm.
By Pythagoras theorem:
Base² + Altitude² = Hypotenuse²
x² + (x - 7)² = 13²
x² + x² - 14x + 49 = 169
2x² - 14x + 49 - 169 = 0
2x² - 14x - 120 = 0
Divide by 2:
x² - 7x - 60 = 0
Factors of -60 summing to -7 are -12 and 5.
x² - 12x + 5x - 60 = 0
x(x - 12) + 5(x - 12) = 0
(x + 5)(x - 12) = 0
x = -5 or x = 12.
Since length cannot be negative, base x = 12 cm.
Altitude = 12 - 7 = 5 cm.
The other two sides are 12 cm and 5 cm.

6. A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was 90, find the number of articles produced and the cost of each article.

Answer:
Let the number of articles produced be x.
Cost of each article = 2x + 3.
Total cost = Number of articles × Cost per article
90 = x(2x + 3)
90 = 2x² + 3x
2x² + 3x - 90 = 0
We need numbers multiplying to 2(-90) = -180 and summing to 3. Numbers are 15 and -12.
2x² + 15x - 12x - 90 = 0
x(2x + 15) - 6(2x + 15) = 0
(x - 6)(2x + 15) = 0
x = 6 or x = -15/2.
Number of articles must be positive integer, so x = 6.
Cost of each article = 2(6) + 3 = 15.
Number of articles = 6, Cost of each article = ₹ 15.

EXERCISE 4.3

1. Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:

(i) 2x² - 3x + 5 = 0

Answer:
Here a = 2, b = -3, c = 5.
Discriminant D = b² - 4ac
D = (-3)² - 4(2)(5)
D = 9 - 40 = -31
Since D < 0, no real roots exist.

(ii) 3x² - 4√3x + 4 = 0

Answer:
Here a = 3, b = -4√3, c = 4.
Discriminant D = b² - 4ac
D = (-4√3)² - 4(3)(4)
D = (16 × 3) - 48 = 48 - 48 = 0
Since D = 0, the equation has two equal real roots.
Roots are given by x = -b/2a
x = -(-4√3) / (2 × 3)
x = 4√3 / 6 = 2√3 / 3
Roots are 2√3/3 and 2√3/3.

(iii) 2x² - 6x + 3 = 0

Answer:
Here a = 2, b = -6, c = 3.
Discriminant D = b² - 4ac
D = (-6)² - 4(2)(3)
D = 36 - 24 = 12
Since D > 0, the equation has two distinct real roots.
x = (-b ± √D) / 2a
x = (6 ± √12) / 4
x = (6 ± 2√3) / 4
x = (3 ± √3) / 2
Roots are (3 + √3)/2 and (3 - √3)/2.

2. Find the values of k for each of the following quadratic equations, so that they have two equal roots.

(i) 2x² + kx + 3 = 0

Answer:
For equal roots, Discriminant D = 0.
b² - 4ac = 0
Here a = 2, b = k, c = 3.
k² - 4(2)(3) = 0
k² - 24 = 0
k² = 24
k = ±√24
k = ±2√6
Values of k are 2√6 and -2√6.

(ii) kx(x - 2) + 6 = 0

Answer:
Rewrite in standard form: kx² - 2kx + 6 = 0
Here a = k, b = -2k, c = 6.
For equal roots, D = 0.
b² - 4ac = 0
(-2k)² - 4(k)(6) = 0
4k² - 24k = 0
4k(k - 6) = 0
So, k = 0 or k = 6.
Since a cannot be 0 for a quadratic equation (coefficient of x²), k ≠ 0.
Therefore, k = 6.

3. Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m²? If so, find its length and breadth.

Answer:
Let breadth = x m.
Length = 2x m.
Area = Length × Breadth
800 = 2x × x
800 = 2x²
x² = 400
x = √400 = 20 (Taking positive value for distance)
Since x is real, it is possible.
Breadth = 20 m.
Length = 2 × 20 = 40 m.
Answer: Yes possible. Length = 40 m, Breadth = 20 m.

4. Is the following situation possible? If so, determine their present ages. The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48.

Answer:
Let the age of one friend be x years.
The age of the other friend = (20 - x) years.
Four years ago:
Age of first friend = x - 4
Age of second friend = (20 - x) - 4 = 16 - x
Product = 48
(x - 4)(16 - x) = 48
16x - x² - 64 + 4x = 48
-x² + 20x - 64 - 48 = 0
-x² + 20x - 112 = 0
x² - 20x + 112 = 0
Check Discriminant D:
D = b² - 4ac
D = (-20)² - 4(1)(112)
D = 400 - 448 = -48
Since D < 0, no real roots exist.
Answer: No, this situation is not possible.

5. Is it possible to design a rectangular park of perimeter 80 m and area 400 m²? If so, find its length and breadth.

Answer:
Perimeter = 2(l + b) = 80 ⇒ l + b = 40 ⇒ b = 40 - l.
Area = l × b = 400
l(40 - l) = 400
40l - l² = 400
l² - 40l + 400 = 0
Check Discriminant:
D = (-40)² - 4(1)(400) = 1600 - 1600 = 0.
Real and equal roots exist, so it is possible.
Solving (l - 20)² = 0
l = 20 m.
Breadth b = 40 - 20 = 20 m.
Answer: Yes possible. Length = 20 m, Breadth = 20 m. (It is a square).

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Quick Review Flashcards - Click to flip and test your knowledge!

Question

What is the standard form of a quadratic equation in the variable $x$?

Answer

$ax^2 + bx + c = 0$, where $a, b, c$ are real numbers and $a \neq 0$.

Question

Which condition must the coefficient $a$ satisfy in the equation $ax^2 + bx + c = 0$ for it to be quadratic?

Answer

$a$ must not be equal to zero ($a \neq 0$).

Question

In the context of polynomial degrees, what defines a quadratic equation?

Answer

An equation of the form $p(x) = 0$ where $p(x)$ is a polynomial of degree 2.

Question

How are the terms of a quadratic polynomial arranged to achieve the standard form of an equation?

Answer

They are written in descending order of their degrees.

Question

Which ancient civilisation is often believed to be the first to solve quadratic equations?

Answer

The Babylonians.

Question

The Babylonian problem of finding two numbers with a given positive sum $p$ and product $q$ is equivalent to solving which equation form?

Answer

$x^2 - px + q = 0$.

Question

Which Greek mathematician developed a geometrical approach for finding lengths that correspond to solutions of quadratic equations?

Answer

Euclid.

Question

Which ancient Indian mathematician provided an explicit formula to solve quadratic equations of the form $ax^2 + bx = c$?

Answer

Brahmagupta ($598 - 665$ C.E.).

Question

To whom is the derivation of the quadratic formula by the method of completing the square credited?

Answer

Sridharacharya ($1025$ C.E.).

Question

Which Arab mathematician studied various types of quadratic equations around $800$ C.E.?

Answer

Al-Khwarizmi.

Question

What is the name of the book published in Europe in $1145$ C.E. by Abraham bar Hiyya Ha-Nasi that gave complete solutions to different quadratic equations?

Answer

'Liber embadorum'.

Question

Under what condition is a real number $\alpha$ considered a root of the quadratic equation $ax^2 + bx + c = 0$?

Answer

If $a\alpha^2 + b\alpha + c = 0$.

Question

What is the relationship between the zeroes of the quadratic polynomial $ax^2 + bx + c$ and the roots of the equation $ax^2 + bx + c = 0$?

Answer

The zeroes of the polynomial and the roots of the equation are the same.

Question

What is the maximum number of roots any quadratic equation can have?

Answer

Two.

Question

In the factorisation method, how are the roots of $ax^2 + bx + c = 0$ found after factorising the polynomial into linear factors?

Answer

By equating each linear factor to zero.

Question

If the breadth of a prayer hall is $x$ and the length is 'one metre more than twice its breadth', what is the expression for the length?

Answer

$(2x + 1)$ metres.

Question

John and Jivanti have $45$ marbles total; if John has $x$ marbles, how many does Jivanti have?

Answer

$45 - x$ marbles.

Question

In Example 1, if John and Jivanti both lose $5$ marbles and the product of their remaining marbles is $124$, what is the resulting quadratic equation?

Answer

$x^2 - 45x + 324 = 0$.

Question

A cottage industry's cost of production for one toy is '$55$ minus the number of toys produced ($x$)'; what is the total cost for $x$ toys?

Answer

$x(55 - x)$.

Question

When checking if an equation like $(x + 2)^3 = x^3 - 4$ is quadratic, what must be done first?

Answer

The equation must be simplified by expanding terms and collecting like terms.

Question

Is the equation $x(x + 1) + 8 = (x + 2)(x - 2)$ a quadratic equation?

Answer

No, because the $x^2$ terms cancel out, leaving a linear equation ($x + 12 = 0$).

Question

What are the roots of the quadratic equation $2x^2 - 5x + 3 = 0$ using factorisation?

Answer

$1$ and $\frac{3}{2}$.

Question

When solving for physical dimensions (like the breadth of a hall), why might one root of a quadratic equation be ignored?

Answer

Dimensions like breadth cannot be negative.

Question

What algebraic method is used to factorise quadratic polynomials by manipulating the $bx$ term?

Answer

Splitting the middle term.

Question

State the Quadratic Formula for finding the roots of $ax^2 + bx + c = 0$.

Answer

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Question

Under what specific condition regarding the discriminant does the Quadratic Formula provide real roots?

Answer

$b^2 - 4ac \geq 0$.

Question

What is the term for the expression $b^2 - 4ac$ in a quadratic equation?

Answer

The discriminant.

Question

Describe the nature of the roots of a quadratic equation if the discriminant is greater than zero ($D > 0$).

Answer

Two distinct real roots.

Question

Describe the nature of the roots of a quadratic equation if the discriminant is equal to zero ($D = 0$).

Answer

Two equal real roots (also called coincident roots).

Question

Describe the nature of the roots of a quadratic equation if the discriminant is less than zero ($D < 0$).

Answer

No real roots.

Question

What is the discriminant of the equation $2x^2 - 4x + 3 = 0$?

Answer

$-8$.

Question

If the discriminant of $2x^2 - 4x + 3 = 0$ is $-8$, what can be concluded about its roots?

Answer

The equation has no real roots.

Question

In a right triangle, if the altitude is $7\text{ cm}$ less than its base $x$, what is the expression for the altitude?

Answer

$(x - 7)\text{ cm}$.

Question

For a right triangle with base $x$, altitude $(x - 7)$, and hypotenuse $13$, which theorem is used to form a quadratic equation?

Answer

Pythagoras theorem ($x^2 + (x - 7)^2 = 13^2$).

Question

What are the roots of the equation $3x^2 - 2x + \frac{1}{3} = 0$?

Answer

$\frac{1}{3}$ and $\frac{1}{3}$ (equal roots).

Question

If two consecutive positive integers are $x$ and $x + 1$, and their product is $306$, what is the quadratic representation?

Answer

$x^2 + x - 306 = 0$.

Question

How is the distance between two diametrically opposite gates $A$ and $B$ of a circular park related to a point $P$ on the boundary?

Answer

Angle $\angle APB$ is $90^\circ$ because it is an angle in a semicircle.

Question

If the distance of a pole from gate $B$ is $x$ and the difference of distances from gates $A$ and $B$ is $7$, what is the distance from gate $A$?

Answer

$(x + 7)$ metres.

Question

When finding values of $k$ for which $2x^2 + kx + 3 = 0$ has equal roots, what value must $b^2 - 4ac$ take?

Answer

Zero ($0$).

Question

If the area of a rectangular plot is $528\text{ m}^2$ and the length is $1$ more than twice the breadth ($x$), what is the equation?

Answer

$2x^2 + x - 528 = 0$.

Question

What is the next step in solving $(x - 12)(2x + 25) = 0$ to find the roots?

Answer

Set each factor to zero: $x - 12 = 0$ or $2x + 25 = 0$.

Question

What value of $x$ is found for the breadth of the prayer hall in Example 6?

Answer

$12\text{ m}$.

Question

If the roots of a quadratic equation are $\alpha$ and $\beta$, and the equation is $x^2 - 45x + 324 = 0$, what is the sum of these roots?

Answer

$45$.

Question

To find the speed of a train that takes $3$ hours more to travel $480\text{ km}$ when speed is reduced by $8\text{ km/h}$, what variable is typically defined as $x$?

Answer

The uniform speed of the train in $\text{km/h}$.

Question

In Example 5, the roots of $3x^2 - 2\sqrt{6}x + 2 = 0$ are both equal to _____.

Answer

$\sqrt{\frac{2}{3}}$.

Question

If a quadratic equation is represented as $x^2 - 55x + 750 = 0$, what does the constant term $750$ represent in the context of the cottage industry example?

Answer

The total cost of production on a particular day.

Question

What is the formula for the roots of $ax^2 + bx + c = 0$ when the discriminant $b^2 - 4ac = 0$?

Answer

$x = -\frac{b}{2a}$.

Question

If the product of the ages of two friends is $48$ and the sum of their present ages is $20$, what equation represents their ages $4$ years ago?

Answer

$(x - 4)(16 - x) = 48$, where $x$ is the present age of one friend.

Question

Is it possible to design a rectangular park with a perimeter of $80\text{ m}$ and an area of $400\text{ m}^2$?

Answer

Yes, it is possible (it would be a square of side $20\text{ m}$).

Question

What is the simplified quadratic form of the equation $(x - 2)^2 + 1 = 2x - 3$?

Answer

$x^2 - 6x + 8 = 0$.