Important Formulas: Quadratic Equations | Quantitative Aptitude for SSC CGL PDF Download

Definition of Quadratic Equation

A quadratic equation is a form of polynomial equation in a single variable, typically denoted as "x," and can be presented in the standard form: ax² + bx + c = 0. In this representation, "a," "b," and "c" are constants, with the condition that "a" must not equal 0.

Formula for Quadratic Equation & Definitions

• An equation where the highest exponent of the variable is a square. Standard form of quadratic equation is ax²+bx+c = 0
• Where,  x is the unknown variable and a, b, c are the numerical coefficients.

Quadratic Equations Formulas

• If ax²+bx+c = 0 is a quadratic equation, then the value of x is given by the following formula
• 

Formula of Quadratic Equation & Method of Quadratic Questions

1. Factorization 

It is very simple method to to solve quadratic equations. Factorization give 2 linear equations

For example:  x² + 3x – 4 = 0
Here, a = 1, b = 3 and c = -4
Now, find two numbers whose product is – 4 and sum is 3.
So, the numbers are 4 and -1.
Therefore, two factors will be 4 and -1

2. Completing the Square Method

Every Quadratic question has always a square term. If we could get two square terms of quality sign we can get a linear equations. Middle term is called as ‘b’ and splited by (b/2)² 
For exampl : – x²+ 4x +4
Here x² =1, b= 4

(x+2)² = 3
Take the square of both side

Therefore, x = -0.27 or -3.73

Formulas of Quadratic Equations & Key points to Remember

Other basic concepts to remember while solving quadratic equations are:

1.Nature of roots

• Nature of roots determine whether the given roots of the equation are real, imaginary, rational or irrational. The basic formula is b² – 4ac.
• This formula is also called discriminant or D. The nature of the roots depends on the value of D. Conditions to determine the nature of the roots are:
• If D < 0, than the given roots are imaginary.
• If D = 0, then roots given are real and equal.
• If D > 0, then roots are real and unequal.
• Also, in case of D > 0, if the equation is a perfect square than the given roots are rational, or else they are irrational.

2. Sum and product of the roots

• For any given equation the sum of the roots will always be  and the product of the roots will be   Thus, the standard quadratic equation can also be written as x² – (Α + Β)x + Α*Β = 0

3. Forming a quadratic equation

• The equation can be formed when the roots of the equation are given or the product and sum of the roots are given.

Questions on Quadratic Equations

Q1: Which of the following is the general form of a quadratic equation?
(a) x² + 3x – 2 = 0
(b) 3x + 2 = 5
(c) 2x – 4 = 0
(d) 5x² – 7x = 3
Ans: (a)
The general form of a quadratic equation is ax² + bx + c = 0, where “a,” “b,” and “c” are constants. Option A follows this format, representing a quadratic equation.

Q2: Which method is most suitable for solving the quadratic equation x² – 6x + 9 = 0?
(a) Factoring
(b) Quadratic Formula
(c) Completing the Square
(d) Graphical Analysis
Ans: (c)
The quadratic equation x² – 6x + 9 = 0 is a perfect square trinomial, and completing the square is the most efficient method to solve it. This method involves transforming the equation into a squared binomial, which makes it easy to find the solutions.

Q3: Which of the following quadratic equations has complex roots?
(a) 2x² + 4x + 5 = 0
(b) x² + 6x + 9 = 0
(c) 3x² – 6x + 3 = 0
(d) 5x² – 10x + 5 = 0
Ans: (a)
The discriminant (Δ) of a quadratic equation determines the nature of its roots. If Δ < 0, the equation has complex roots. For option A, the discriminant is 4² – 4(2)(5) = 16 – 40 = -24, which is negative. Therefore, the quadratic equation 2x² + 4x + 5 = 0 has complex roots.

Q4: What is the discriminant of the quadratic equation 2x² – 5x + 3 = 0?
(a) 1
(b) -11
(c) -19
(d) 11
Ans: (c)
The discriminant (Δ) of a quadratic equation ax² + bx + c = 0 is given by the expression Δ = b² – 4ac. Substituting the coefficients from the given equation, we have Δ = (-5)² – 4(2)(3) = 25 – 24 = 1. Therefore, the discriminant is 1.

Q5: For the quadratic equation 2x² + 5x – 3 = 0, what are the roots?
(a) x = -3 and x = 1/2
(b) x = 3 and x = -1/2
(c) x = -3 and x = -1/2
(d) x = 3 and x = 1/2
Ans: (a)
The quadratic equation can be factored as (2x – 3)(x + 1) = 0. Setting each factor equal to zero gives 2x – 3 = 0 and x + 1 = 0. Solving for “x” in each equation yields x = 3/2 and x = -1. Therefore, the roots are x = -3 and x = 1/2.