Solved Questions on Quadratic Equations
1. What is the factorization of the quadratic equation 2x2 + 7x + 6?
a) (x + 1) (2x + 3)
b) (2x + 1) (x + 3)
c) (x + 2) (2x + 3)
d) (2x + 2) (x + 3)
Answer: c) (x + 2) (2x + 3)
Explanation: The factors are obtained by finding two numbers whose product is equal to the product of the coefficient of x^2 and the constant term, and whose sum is equal to the coefficient of x.
2x2 + 7x + 6
2x2 + 3x + 4x + 6
x (2x + 3) + 2(2x + 3)
(x + 2) (2x + 3)
2. What is the factorization of the quadratic equation 4y2 – 12y + 9?
a) (2y – 3) (2y – 3)
b) (2y – 3)2
c) (2y – 3) (2y + 3)
d) (2y + 3)2
Answer: b) (2y – 3)2
Explanation: The equation is a perfect square trinomial, Hence
4y2 – 12y + 9 = (2y)2 – 2 x 2y x 3 + 3²
Since (a – b)2 = a² - 2 x a x b + b²
= (2y – 3)2
3. What are the roots of the quadratic equation 2x2 + 3x – 5 = 0 using the quadratic formula?
a) x = 1 and -5/2
b) x = (3 ± √ (29)) / 4
c) x = (-3 ± √ (29)) / 2
d) x = -3/2 and 2 and
Answer: a) x = 1 and -5/2
Explanation: By substituting the values of a, b, and c into the quadratic formula, we can calculate the roots of the equation.
X = [(-3) ± √(9 + 40)]/4
x = [ -3 ± √ 49] / 4
x = [ -3 ± 7] / 4
x = 1 and -5/2
4. What is the discriminant of the quadratic equation 3x2 – 5x + 2 = 0?
a) 1
b) -1
c) 0
d) 9
Answer: c) 0
Explanation: The discriminant is calculated as b2 – 4ac. If the discriminant is equal to zero, it indicates that the equation has two identical roots.
B2 – 4ac = 25 – 4 x 3 x 2
= 25 – 24 = 1
5. Given equations (I) and (II), solve both equations, comparing them and select the appropriate answer from the provided options.
I) x2 = 49
II) y2 + 15y + 56 = 0
a) x > y
b) x < y
c) x ≥ y
d) x ≤ y
Answer: c) x ≥ y
Explanation: The factor of x2 = 49 are ± 7 and factors of y2 + 15y + 56 = 0 are -7 and -8. Therefore, the appropriate answer is x ≥ y.
Worksheet for Quadratic Equations
1. What is the standard form of a quadratic equation?
a) ax2 + bx + c = 0
b) ax2 + c = bx
c) ax + bx2 + c = 0
d) a + bx2 + cx = 0
Answer: a) ax2 + bx + c = 0.
2. How many solutions does a quadratic equation have?
a) One
b) Two
c) Three
d) Four
Answer: b) Two.
3. What is the nature of the roots of a quadratic equation x² - 20x +91?
a) Real and equal
b) Real and distinct
c) Imaginary roots
d) Complex roots
Answer: b) Real and distinct
4. What is the factorization of the quadratic equation 5x2 + 4x - 1?
a) (5x - 1) (x + 1)
b) (5x + 1) (x - 1)
c) (5x - 1) (x - 1)
d) (5x + 1) (x + 1)
Answer: a) (5x - 1) (x + 1)
5. What is the factorization of the quadratic equation x2 - 2x - 3?
a) (x - 3) (x - 1)
b) (x + 3) (x + 1)
c) (x - 3) (x + 1)
d) (x + 3) (x - 1)
Answer: c) (x - 3) (x + 1)
6. What is the factorization of the quadratic equation 3x2 - 10x + 4?
a) (3x - 2) (x - 2)
b) (3x + 2) (x + 2)
c) (3x - 4) (x + 1)
d) (3x + 4) (x - 1)
Answer: a) (3x - 2) (x - 2)
7. What is the factorization of the quadratic equation x2 - 7x + 12?
a) (x - 3) (x - 4)
b)(x + 3) (x + 4)
c)(x - 3) (x + 4)
d)(x + 3) (x - 4)
Answer: a) (x - 3) (x - 4)
8. What is the factorization of the quadratic equation 2x2 + x - 3?
a) (2x + 3) (x - 1)
b) (2x - 3) (x + 1)
c) (2x + 1) (x - 3)
d) (2x - 1) (x + 3)
Answer: a) (2x + 3) (x - 1)
9. What is the factorization of the quadratic expression 25x2 - 20x + 4 using the completing square method?
a) (5x - 2) (5x - 2)
b) (5x + 2) (5x + 2)
c) (5x - 4) (5x - 4)
d) (5x - 6) (5x - 6)
Answer: a) (5x - 2) (5x - 2)
10. Given equations (I) and (II), solve both equations, comparing them and select the appropriate answer from the provided options.
(I) 2x - 5 = 13
(II) 3y + 7 = -2y + 1
a) x > y
b) x < y
c) x ≥ y
d) x ≤ y
Answer: a) x > y
