## Basic Concepts of Quadratic Equations

A Quadratic Equation is the equation that can be rearranged in standard form ax2 + bx + c = 0 as where x is a variable and a, b, and c represent constants , where a ≠ 0. If a = 0, then the equation is linear, not quadratic, as there is no term. Let us have a look on some of the basic concepts and Formula for Quadratic Equation that will help you to memorize this topic easily and rapidly. ### 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.

• If ax2+bx+c = 0 is a quadratic equation, then the value of x is given by the following formula
• x = $\frac{-b \pm \sqrt{b^2 – 4ac}}{2a}$

• 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

• 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 $(\frac{b}{2})^2$

For example :  – x²+ 4x +4

Here x² =1, b= 4

$(\frac{b}{2})^2$ = $(\frac{4}{2})^2$ = 4

x²+4x+4 = -1+4

(x+2)² = 3

Take the square of both side

x+2 = ±$\sqrt{3}$ = ± 1.73

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  $– \frac{b}{a}$, and the product of the roots will be  $– \frac{c}{a}$. Thus, the standard quadratic equation can also be written as x2 – (Α + Β)x + Α*Β = 0

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

Question 1:

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

Explanation:

The correct answer is option 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.

Question 2:

What is the discriminant of the quadratic equation 2x² – 5x + 3 = 0?

A) 1

B) -11

C) -19

D) 11

Explanation:

The correct answer is option 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.

Question 3:

Which method is most suitable for solving the quadratic equation x² – 6x + 9 = 0?

A) Factoring

C) Completing the Square

D) Graphical Analysis

Explanation:

The correct answer is option 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.

Question 4:

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

Explanation:

The correct answer is option 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.

Question 5:

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

Explanation:

The correct answer is option 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.

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