Quadratic Equations are polynomial equations of degree 2 in one variable, such as $$f(x) = ax^2 + bx + c = 0, $$where a, b, and c are all in R and a is not zero. It is the general form of a quadratic equation, with ‘a’ denoting the leading coefficient and ‘c’ denoting the absolute term of f (x). The quadratic equation’s roots are the values of x that satisfy the equation (α, β).
Quadratic Equation Formula
The quadratic formula determines the solution or roots of a quadratic problem.
$$ x = (α, β) = \frac{b±\sqrt {b^24ac }}{2a} $$
How do you solve quadratic equations?
There are generally four ways to solve quadratic problems. They are:
 Factoring
 Complete the square.
 Using the Quadratic Formula
 Taking the Square Root
Practice Quizzes
Quadratic equations Quiz 1 – Coming Soon 
Quadratic equations Quiz 2 – Coming Soon  Quadratic equations Quiz 3 – Coming Soon 
Solved Quadratic Equations
Q1. (i) 2x² + 9x + 9 = 0
(ii) 15y² + 16y + 4 = 0
(a) x > y
(b) x < y
(c) x ≥ y
(d) x ≤ y
(e) x = y or no relation can be established between x & y.
Ans.(b)
Sol.
(i) 2x² + 9x + 9 = 0
2x² + (6 + 3) x + 9 = 0
2x (x + 3) + 3 (x + 3) = 0
x = –3/2 , –3
(ii) 15y² + 16y + 4 = 0
15y² + 10y + 6y + 4 = 0
5y (3y + 2) + 2 (3y + 2) = 0
y = –2/5, –2/3
x < y
Q2. (i) 2x³ = √256
(ii) 2y² – 9y + 10 = 0
(a) x = y or no relation can be established between x & y.
(b) x < y
(c) x ≤ y
(d) x ≥ y
(e) x > y
Ans.(c)
Sol.
(i) 2x³ = 16
x³ = 8
x = 2
(ii) 2y² – 9y + 10 = 0
2y² – (5 + 4) y + 10 = 0
2y² – 5y – 4y + 10 = 0
y (2y – 5) – 2 (2y – 5) = 0
y = 2, 5/2
x ≤ y
Q3. (i) 6x² – 11x + 4 = 0
(ii) 3y² – 5y + 2 = 0
(a) x ≤ y
(b) x < y
(c) x ≥ y
(d) x > y
(e) x = y or no relation can be established between x & y.
Ans.(e)
Sol.
(i) 6x² – 11x + 4 = 0
6x² – (8 + 3) x + 4 = 0
6x² – 8x – 3x + 4 = 0
2x (3x – 4) – 1 (3x – 4) = 0
x = 12, 43
(ii) 3y² – 5y + 2 = 0
3y² – (3 + 2) y + 2 = 0
3y² – 3y – 2y + 2 = 0
3y (y – 1) – 2 (y – 1) = 0
y = ⅔, 1
No relation between x and y
Q4. (i) 3x² + 11x + 10 = 0
(ii) 2y² + 11y + 14 = 0
(a) x ≥ y
(b) x ≤ y
(c) x > y
(d) x < y
(e) x = y or no relation can be established between x & y.
Ans.(a)
Sol.
(i) 3x² + 11x + 10 = 0
3x² + 6x + 5x + 10 = 0
3x (x + 2) + 5 (x + 2) = 0
x = – 2, –5/3
(ii) 2y² + 11y + 14 = 0
2y² + 7y + 4y + 14 = 0
y (2y + 7) +2 (2y + 7) = 0
y = –2, –7/2
x ≥ y
Q5. (i) 12x² + 11x + 2 = 0
(ii) 12y²+ 7y + 1 = 0
(a) x ≥ y
(b) x = y or no relation can be established between x & y.
(c) x < y
(d) x ≤ y
(e) x > y
Ans.(b)
Sol.
(i) 12x² + 8x + 3x + 2 = 0
4x (3x + 2) + 1 (3x + 2) = 0
x = –2/3,–1/4
(ii) 12y² + 7y + 1 = 0
12y² + 4y + 3y + 1 = 0
4y (3y + 1) +1 (3y + 1) = 0
y = –1/3,–1/4
No relation between x and y
Q6. (i) 21x² + 10x + 1 = 0
(ii) 24y²+ 26y + 5 = 0
(a) x ≤ y
(b) x = y or no relation can be established between x & y.
(c) x ≥ y
(d) x > y
(e) x < y
Ans.(b)
Sol.
(i) 21x² + 10x + 1 = 0
21x² + 7x + 3x + 1 = 0
7x (3x + 1) + 1 (3x + 1) = 0
x = –1/3, –1/7
(ii) 24y² + 26y + 5 = 0
24y² + (20 + 6)y + 5 = 0
24y² + 20y + 6y + 5 = 0
4y (6y + 5) + 1 (6y + 5) = 0
y = –5/6, –1/4
No relation between x and y.
Q7.
I. x² – 7x + 12 = 0
II. y² – 8y + 12 = 0
(a) If x > y
(b) If x ≥ y
(c) If y > x
(d) If y ≥ x
(e) If x = y or no relation can be established
Ans.(e)
Sol.
I. x² – 7x + 12 = 0
x²−4𝑥−3𝑥+12=0
(𝑥−4)(𝑥−3)=0 𝑥=3,4
II. y² – 8y + 12 = 0
y²−6𝑦−2𝑦+12=0
(𝑦−6)(𝑦−2)=0 𝑦=2,6
No relation can be established
Q8.
I. 2x² + x – 28 = 0
II. 2y² – 23y + 56 = 0
(a) If x > y
(b) If x ≥ y
(c) If y > x
(d) If y ≥ x
(e) If x = y or no relation can be established
Ans.(d)
Sol.
I. 2x² + x – 28 = 0
2x² + 8x – 7x – 28 = 0
2x (x + 4) – 7 (x + 4) = 0
(2x – 7) (x + 4)= 0
𝑥=−4,7/2
II. 2y² – 23y + 56 = 0
2y² – 16y – 7y + 56 = 0
2y(y – 8) – 7(y – 8) = 0
(2y – 7) (y – 8) = 0
𝑦=7/2,8
y ≥ x
Q9.
I. 2x² – 7x – 60 = 0
II. 3y² + 13y + 4 = 0
(a) If x > y
(b) If x ≥ y
(c) If y > x
(d) If y ≥ x
(e) If x = y or no relation can be established
Ans.(e)
Sol.
I. 2x² – 7x – 60 = 0
2x² – 15x + 8x – 60 = 0
x (2x – 15 ) + 4 (2x – 15) = 0
(x + 4) (2𝑥−15)=0
𝑥= −4,15/2
II. 3y² + 13y + 4 = 0
3y² + 12y + y + 4 = 0
3y (y + 4) + 1 (y + 4) = 0
(3y + 1) (y + 4) = 0
𝑦=−13,−4
No relation between x and y
Q10.
I. x² – 17x – 84 = 0
II. y² + 4y – 117 = 0
(a) If x > y
(b) If x ≥ y
(c) If y > x
(d) If y ≥ x
(e) If x = y or no relation can be established
Ans.(e)
Sol.
I. x² – 17x – 84 = 0
x² +4x – 21x – 84 = 0
(x + 4) (x – 21) = 0
x = 4, 21
II. y² + 4y – 117 = 0
y² – 9y + 13y – 117 = 0
(y – 9) (y + 13) = 0
y = 9, 13
No relation between x and y
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Frequently Asked Questions About Quadratics
Q1.What is a quadratic equation?
A quadratic equation is a polynomial equation with the maximum degree equal to two. The equation is provided by ax² + bx + c = 0, where an is not zero.
Q2: What are the procedures for solving a quadratic equation?
There are four main approaches for solving quadratic equations. They are Factorisation, Using Square Roots, Complete the square and Using Quadratic Formula.
Q3.Write the quadratic equation as a sum and product of roots.
If α and β are the roots of a quadratic equation, their total is α+β.
The product of roots equals αβ.
Thus, the required equation is: x^2 – (α+β)x + (αβ) = 0.
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