Mathematics Classes VI-X for School Assistants and TET Paper 2A · Chapter 4

Quadratic Equations

What to remember

A quadratic equation is ax² + bx + c = 0 with a ≠ 0. It has at most two roots. A root is a value of x that makes the equation true; it is the same as a zero of the polynomial ax² + bx + c.
The quadratic formula is x = (−b ± √(b² − 4ac)) / 2a. The quantity D = b² − 4ac is the discriminant. It tells the nature of the roots without solving the equation.
Sum of roots = −b/a and product of roots = c/a. These two results solve many problems about forming equations and finding unknown constants.

Standard form and roots

A quadratic equation in the variable x has the standard form ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0. If a = 0 the equation becomes linear. The values of x that satisfy the equation are its roots (solutions).

Is an equation quadratic? Rewrite it in standard form first. For example, (x + 1)² = 2(x − 3) becomes x² + 2x + 1 = 2x − 6, so x² + 7 = 0, which is quadratic. But x(x + 1) + 8 = (x + 2)(x − 2) becomes x² + x + 8 = x² − 4, so x + 12 = 0, which is linear and not quadratic. Always expand and simplify before deciding.

A number k is a root if putting x = k satisfies the equation. Example: Is 3 a root of x² − 5x + 6 = 0? 9 − 15 + 6 = 0, so yes.

Solving by factorisation

Split the middle term bx into two terms whose product is ac and whose sum is b. Then take out common factors and use the rule: if p × q = 0 then p = 0 or q = 0.

Example 1: x² − 5x + 6 = 0. Two numbers with product 6 and sum −5 are −2 and −3. So (x − 2)(x − 3) = 0 and x = 2 or x = 3.

Example 2: 2x² − 7x + 3 = 0. ac = 6; numbers with product 6 and sum −7 are −1 and −6. So 2x² − x − 6x + 3 = 0, which gives x(2x − 1) − 3(2x − 1) = 0, so (2x − 1)(x − 3) = 0 and x = 1/2 or x = 3.

Solving by completing the square

Divide the equation by a, move the constant to the right side, and add (half the coefficient of x)² to both sides.

Example 3: x² + 4x − 5 = 0. Then x² + 4x = 5; add 4 on both sides: (x + 2)² = 9. So x + 2 = ±3, giving x = 1 or x = −5.

This method is the base of the quadratic formula. Starting from ax² + bx + c = 0 and completing the square leads to x = (−b ± √(b² − 4ac)) / 2a. The formula is traditionally credited to the Indian mathematician Sridharacharya.

Quadratic formula and the discriminant

For ax² + bx + c = 0 the roots are α = (−b + √D)/2a and β = (−b − √D)/2a, where D = b² − 4ac.

Value of D	Nature of roots	Graph of y = ax² + bx + c
D > 0	Two distinct real roots	Cuts the x-axis at two points
D = 0	Two equal real roots, each −b/2a	Touches the x-axis at one point
D < 0	No real roots	Does not meet the x-axis

If a, b, c are rational and D is a perfect square, the roots are rational. If a, b, c are rational and D is positive but not a perfect square, the roots are irrational and come as a pair of conjugates.

Example 4: x² − 4x − 1 = 0. D = 16 + 4 = 20, so x = (4 ± √20)/2 = 2 ± √5.

Example 5: 2x² − 4x + 3 = 0 has D = 16 − 24 = −8 < 0, so no real roots.

Example 6: Find k for equal roots in kx(x − 2) + 6 = 0. In standard form kx² − 2kx + 6 = 0. D = 4k² − 24k = 0 gives 4k(k − 6) = 0. Since k ≠ 0 for a quadratic, k = 6.

Relations between roots and coefficients

If α and β are the roots of ax² + bx + c = 0:

Quantity	Value
α + β	−b/a
αβ	c/a
α − β	±√D / a
α² + β²	(α + β)² − 2αβ
1/α + 1/β	(α + β)/αβ = −b/c

To form an equation with given roots: x² − (sum)x + (product) = 0. For roots 3 and −2: x² − x − 6 = 0. If one root is the reciprocal of the other, then c = a. If the roots are equal in size and opposite in sign, then b = 0. If one root is zero, c = 0.

Least and greatest value

The graph of y = ax² + bx + c has its turning point (vertex) at x = −b/2a. If a > 0 the expression has a least value there, equal to (4ac − b²)/4a. If a < 0 it has a greatest value of the same form. Example: x² − 6x + 11 has its least value at x = 3, and the value is 9 − 18 + 11 = 2.

Equations that reduce to quadratics

Some equations become quadratic after a substitution. For x⁴ − 5x² + 4 = 0 put y = x²; then y² − 5y + 4 = 0, so y = 1 or 4, and x = ±1 or ±2. For x + 1/x = 5/2 multiply by x to get 2x² − 5x + 2 = 0, so x = 2 or 1/2. Always check that the answers do not make a denominator zero.

Word problems

Steps: choose a variable, translate the sentence into an equation, solve it, discard answers that do not suit the situation (for example, negative lengths or ages), and check.

Example 7: The product of two consecutive positive integers is 306. If the smaller is n, then n(n + 1) = 306, so n² + n − 306 = 0, which factors as (n − 17)(n + 18) = 0. So n = 17 and the integers are 17 and 18. (The root −18 is dropped.)

Example 8: A rectangle has perimeter 26 m and area 40 m². With one side x, the other is 13 − x, so x(13 − x) = 40, which gives x² − 13x + 40 = 0, so (x − 5)(x − 8) = 0. The sides are 5 m and 8 m.

Example 9: A train covers 360 km at a uniform speed. If the speed were 5 km/h more, the journey would take 1 hour less. Let the speed be x km/h: 360/x − 360/(x + 5) = 1. This gives x² + 5x − 1800 = 0. D = 25 + 7200 = 7225 = 85², so x = (−5 + 85)/2 = 40 km/h.

Classroom angle

Let students check several roots by substitution first so they see what a root is. Teach all three methods and show that they agree for one equation. A frequent mistake is to cancel a common factor of x on both sides and so lose the root x = 0. For example, in x² = 3x the correct step is x(x − 3) = 0, which gives x = 0 or 3. Another is to forget the ± sign in the formula or to mis-handle signs in b² − 4ac when b is negative.

Exam traps

a = 0: the equation is then linear, not quadratic.
Cancelling x: dividing by x loses the root x = 0.
The sign of b: in −b ± √D, the b takes its own sign; if b = −5 then −b = 5.
D = 0: two equal roots (one repeated root); this is not the same as "no root".
D < 0: no real roots, but this does not mean the equation has no solution in complex numbers.
Sum and product signs: sum = −b/a, product = +c/a.
Word problems: reject negative values for length, age and number of objects.
Equation forms: (x + 1)² = 2(x − 3) is quadratic; x(x + 1) + 8 = (x + 2)(x − 2) is not.

One-liners

1. The standard form is ax² + bx + c = 0 with a ≠ 0.
2. A quadratic equation has at most two roots.
3. The quadratic formula is x = (−b ± √(b² − 4ac)) / 2a.
4. The discriminant is D = b² − 4ac.
5. D > 0: two distinct real roots.
6. D = 0: two equal real roots.
7. D < 0: no real roots.
8. Sum of roots = −b/a; product of roots = c/a.
9. An equation with given roots is x² − (sum)x + product = 0.
10. Equal roots have the value −b/2a.
11. Completing the square adds (half the coefficient of x)² to both sides.
12. The graph of a quadratic is a parabola.

Practice questions

In the standard form ax² + bx + c = 0 of a quadratic equation, which condition must hold?
1. a = b
2. b ≠ 0
3. a ≠ 0
4. c ≠ 0
Answer
C. a ≠ 0
If a = 0 the equation becomes linear.
Which of the following is a quadratic equation?
1. x(x + 1) + 8 = (x + 2)(x − 2)
2. x³ − 4x = 0
3. (x + 1)² = 2(x − 3)
4. 3x + 7 = 0
Answer
C. (x + 1)² = 2(x − 3)
(x + 1)² = 2(x − 3) simplifies to x² + 7 = 0; the second simplifies to x + 12 = 0 (linear).
The roots of x² − 7x + 12 = 0 are
1. −3 and −4
2. 2 and 6
3. 3 and 4
4. 1 and 12
Answer
C. 3 and 4
x² − 7x + 12 = (x − 3)(x − 4).
The discriminant of ax² + bx + c = 0 is
1. −b ± √(b² − 4ac)
2. b² + 4ac
3. 4ac − b
4. b² − 4ac
Answer
D. b² − 4ac
D = b² − 4ac.
The discriminant of 2x² − 4x + 3 = 0 is
1. 8
2. −8
3. 40
4. −4
Answer
B. −8
D = 16 − 24 = −8, so there are no real roots.
If the discriminant of a quadratic equation is zero, the roots are
1. irrational and distinct
2. real and distinct
3. not real
4. real and equal
Answer
D. real and equal
D = 0 gives a repeated root −b/2a.
The roots of 2x² − 7x + 3 = 0 are
1. −1/2 and −3
2. 2 and 3/2
3. 1/2 and 3
4. 1 and 3
Answer
C. 1/2 and 3
(2x − 1)(x − 3) = 0.
The roots of x² + 4x − 5 = 0 are
1. −1 and −5
2. 1 and −5
3. −1 and 5
4. 1 and 5
Answer
B. 1 and −5
(x − 1)(x + 5) = 0.
The roots of x² − 4x − 1 = 0 are
1. 2 ± √5
2. −2 ± √5
3. 4 ± √5
4. 2 ± √3
Answer
A. 2 ± √5
x = (4 ± √20)/2 = 2 ± √5.
The sum of the roots of 3x² − 9x + 2 = 0 is
1. −3
2. 3
3. 2/3
4. 9
Answer
B. 3
Sum = −b/a = 9/3 = 3.
The product of the roots of x² − x − 6 = 0 is
1. 6
2. −1
3. −6
4. 1
Answer
C. −6
Product = c/a = −6.
A quadratic equation with roots 3 and −2 is
1. x² − 5x − 6 = 0
2. x² − x + 6 = 0
3. x² + x − 6 = 0
4. x² − x − 6 = 0
Answer
D. x² − x − 6 = 0
Sum = 1 and product = −6, so x² − x − 6.
The value of k for which kx(x − 2) + 6 = 0 has two equal roots is
1. 6
2. −6
3. 4
4. 2
Answer
A. 6
kx² − 2kx + 6 = 0 has D = 4k² − 24k = 0, so k = 6.
The values of k for which 2x² + kx + 3 = 0 has equal roots are
1. ±12
2. ±√6
3. ±6
4. ±2√6
Answer
D. ±2√6
D = k² − 24 = 0, so k = ±2√6.
The solutions of x² = 3x are
1. 3 only
2. 0 only
3. 0 and 3
4. −3 and 3
Answer
C. 0 and 3
x(x − 3) = 0 gives both roots; dividing by x would lose x = 0.
The product of two consecutive positive integers is 306. The smaller integer is
1. 16
2. 17
3. 18
4. 19
Answer
B. 17
n(n + 1) = 306 gives n = 17.
A rectangle has perimeter 26 m and area 40 m². Its longer side is
1. 8 m
2. 13 m
3. 6 m
4. 10 m
Answer
A. 8 m
Sides satisfy x(13 − x) = 40, so 5 and 8.
A train covers 360 km at a uniform speed. If the speed were 5 km/h more, it would take 1 hour less. The speed is
1. 45 km/h
2. 30 km/h
3. 35 km/h
4. 40 km/h
Answer
D. 40 km/h
360/x − 360/(x + 5) = 1 gives x² + 5x − 1800 = 0, so x = 40.
If −2 is a root of x² + kx − 10 = 0, then k is
1. 3
2. −7
3. −3
4. 7
Answer
C. −3
4 − 2k − 10 = 0, so k = −3.
The equation x² + 1 = 0 has
1. exactly one real root
2. no real roots
3. two equal real roots
4. two distinct real roots
Answer
B. no real roots
D = −4 < 0.
The least value of x² − 6x + 11 is
1. 2
2. 11
3. 3
4. −2
Answer
A. 2
Vertex at x = 3: 9 − 18 + 11 = 2.
The number of real roots of x⁴ − 5x² + 4 = 0 is
1. 2
2. 4
3. 3
4. 0
Answer
B. 4
y = x² gives y = 1 or 4, so x = ±1 and ±2.
The roots of x + 1/x = 5/2 are
1. 5 and 1/2
2. 1 and 3/2
3. 2 and −1/2
4. 2 and 1/2
Answer
D. 2 and 1/2
2x² − 5x + 2 = 0 gives (2x − 1)(x − 2) = 0.
The value of k for which x² − 6x + k = 0 has equal roots is
1. 6
2. 3
3. 36
4. 9
Answer
D. 9
D = 36 − 4k = 0, so k = 9.
If α and β are the roots of x² − 3x + 2 = 0, then α² + β² is
1. 9
2. 7
3. 5
4. 13
Answer
C. 5
(α + β)² − 2αβ = 9 − 4 = 5.
If α and β are the roots of x² − 7x + 10 = 0, then 1/α + 1/β is
1. 10/7
2. 7/10
3. 3
4. 17/10
Answer
B. 7/10
(α + β)/αβ = 7/10.
The discriminant of x² − 6x + 9 = 0 is
1. 0
2. −36
3. 36
4. 18
Answer
A. 0
D = 36 − 36 = 0.
If the roots of ax² + bx + c = 0 are reciprocals of each other, then
1. a = −c
2. b = 0
3. c = a
4. c = 0
Answer
C. c = a
Product of roots = c/a = 1.
If the roots of ax² + bx + c = 0 are equal in size and opposite in sign, then
1. b = 0
2. c = 0
3. a = c
4. a = 0
Answer
A. b = 0
Sum of roots = −b/a = 0.
The quadratic formula is traditionally credited to
1. Pingala
2. Sridharacharya
3. Aryabhata II
4. Euclid
Answer
B. Sridharacharya
The formula is traditionally linked with the Indian mathematician Sridharacharya.
To complete the square in x² + 8x = 20, we add which number to both sides?
1. 8
2. 64
3. 16
4. 4
Answer
C. 16
(half of 8)² = 16.
The roots of 3x² − 5x + 2 = 0 are
1. −1 and −2/3
2. 2 and 1/3
3. 1 and 3/2
4. 1 and 2/3
Answer
D. 1 and 2/3
(x − 1)(3x − 2) = 0.
If x² − 7x + 12 = 0 has roots α > β, then α − β is
1. 1
2. 7
3. 12
4. 5
Answer
A. 1
The roots are 4 and 3.
If one root of x² − 6x + k = 0 is twice the other, then k is
1. 12
2. 9
3. 6
4. 8
Answer
D. 8
Roots r and 2r give 3r = 6, r = 2, so roots 2, 4 and k = 8.
The sum of the squares of two consecutive positive integers is 61. The integers are
1. 4 and 5
2. 6 and 7
3. 5 and 6
4. 3 and 4
Answer
C. 5 and 6
n² + (n + 1)² = 61 gives n² + n − 30 = 0, so n = 5.
Consider the statements: 1. If D > 0, the equation has two distinct real roots. 2. If D < 0, the equation has two equal real roots.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
For D < 0 there are no real roots.
Consider the statements: 1. If a = 0, ax² + bx + c = 0 is still quadratic. 2. The product of the roots is −c/a.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
D. Neither 1 nor 2
If a = 0 it is linear; the product of the roots is c/a.
Consider the statements: 1. A quadratic equation has at most two real roots. 2. The graph of a quadratic polynomial is a parabola.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Both are standard facts.
Consider the statements: 1. Dividing x² = 3x by x gives all its roots. 2. The equation x² = 3x has roots 0 and 3.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
Dividing by x loses the root 0.
Consider the statements: 1. When D = 0 the root is −b/2a. 2. When D = 0 the parabola touches the x-axis at one point.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Both describe the repeated root case.
Consider the statements: 1. If D < 0 the graph of the quadratic does not meet the x-axis. 2. If D > 0 the roots are always rational.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
If D > 0 but not a perfect square the roots are irrational.
Consider the statements: 1. The roots of x² + 1 = 0 are real. 2. x² − 4 = 0 has two real roots.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
x² + 1 = 0 has D < 0.
Consider the statements: 1. In a word problem on length, a negative root is rejected. 2. A negative root is invalid in every word problem.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
A negative value may be valid in some situations, such as temperature.
The roots of x² − 25 = 0 are
1. 5 only
2. 25 and −25
3. 0 and 5
4. 5 and −5
Answer
D. 5 and −5
x² = 25 gives x = ±5.
The roots of x² − 2x + 1 = 0 are
1. 1 and 1
2. 1 and −1
3. 2 and 1
4. 0 and 2
Answer
A. 1 and 1
(x − 1)² = 0, a repeated root.

Page 1 of 1

Chapter quiz · 10 questions

0/10 answered

On what you just read, from the live question bank. 🔒 Answers and solutions are shown to registered students (free).
अध्याय पढ़ने के बाद अभ्यास करें।

1.Let a be an odd positive integer. If the roots of a quadratic equation x² − (a − 1)x + 4a + 7 = 0 are integers, then the least value of a is:
2.If both the roots of the quadratic equation A(8x2 + 4) + Bx + 4x2 − 8 = 0 and 3A(2x2 + 1) + Kx + 3x2 − 6 = 0 are common, then what is the value of 9B2 − 16K2?
3.What is the nature of the roots of the following quadratic equation?
4.Two quadratic equations x² + x + p = 0 and x² + 2x − 3 = 0 have one root common. What is a possible value of p?
5.If the roots of the quadratic equation kx2 − 4x + 1 = 0 are real and equal, what is the value of k?
6.Find the discriminant of the quadratic equation 3x2 − 4x + 2 = 0, and hence determine the nature of its roots.
7.Reduce the equation x4 − 13x2 + 36 = 0 into a quadratic equation and find the roots of the reduced quadratic equation.
8.Two quadratic equations, x² - 5x + 6 = 0 and 2x2 - kx + 6 = 0, have one common root. What is the possible value
of k?
9.Two quadratic equations, 2x2 + 8x + 6 = 0 and kx2 + 4kx + 3 = 0, have both roots common. What is the value of k?
10.The quadratic equation whose roots are 9 and 11 is:

10 left. You can check any time.