Short Notes: Quadratic Equations

What is Quadratic Equation?

• A quadratic polynomial of the form ax² + bx + c, where a ≠ 0 and a, b, c are real numbers, is called a quadratic equation when ax² + bx + c = 0.

• Here a and b are the coefficients of x² and x respectively and 'c' is a constant term.
• Any value is a solution of a quadratic equation if and only if it satisfies the quadratic equation.

How to Find the Solution of a Quadratic Equation by Factorisation?

For a quadratic equation, a real number α is called the root of a quadratic equation ax² + bx + c = 0. Hence, we can write aα² + bα + c = 0. So, x= α is the solution of a quadratic equation or the root of a quadratic equation. In other words, α satisfies the given quadratic equation.

Note: The zeros of the quadratic equation ax²+bx+c = 0 are the same as the root of the quadratic equation ax²+bx+c = 0.

Example: Solve the quadratic equation 2x²+x-300 = 0 by the factorisation method.

Solution: 

Given quadratic equation: 2x²+x-300 = 0

By using factorisation, the quadratic equation 2x² + x -300 = 0 is written as:

2x² - 24x+25x -300 = 0

2x(x-12) +25(x-12) =0

(i.e) (x-12)(2x+25) = 0

Therefore, x-12=0 and 2x+25 = 0

x-12 = 0

Therefore, x= 12.

Similarly, 2x+25 = 0

2x= -25

x =-25/2

x = -12.5.

Hence, the roots of the quadratic equation 2x²+x-300 = 0 are 12 and -12.5.

Quadratic Formula 

• The roots, if a quadratic equation ax² + bx + c = 0 are given by:

• Here, the value b² - 4ac is known as the discriminant and is generally denoted by D. 'D' helps us to determine the nature of roots for a given quadratic equation. Thus D = b² - 4ac.

Example: Find the roots of quadratic equation x² - 7x + 10 = 0 using quadratic formula.

Solution: 

Here, a = 1, b = -7 and c = 10. Then by quadratic formula:

Therefore, x = 2, x = 5.

Nature of Roots of a Quadratic Equation

• If D = 0 ⇒ The roots are Real and Equal.
• If D > 0 ⇒ The two roots are Real and Unequal.
• If D < 0 ⇒ No Real roots exist.

Example: Discuss the nature of the roots of the quadratic equation 2x² - 8x + 3 = 0.

Solution: Here the coefficients are all rational. The discriminant D of the given equation is

D = b² - 4ac = (-8)² - 4 x 2 x 3

= 64 - 24

= 40 > 0

Clearly, the discriminant of the given quadratic equation is positive but not a perfect square. Therefore, the roots of the given quadratic equation are real, irrational and unequal.

Relationship between roots and Coefficients of Quadratic Equation

If α and β are the roots of the quadratic equation, then Quadratic equation is 

x² - (α + β) x + αβ = 0 

OR

 x² - (sum of roots) x + product of roots = 0

where,

• Sum of roots (α + β) =
• Product of roots (α x β) =

Example: If α and β are the roots of the equation x² - 4x + 2 = 0, find the value of

i) α² + β²

ii) α² - β²

iii) α³ - β³

^{iv)1/α  + 1/ β}

^Solution:

The given equation is x² - 4x + 2 = 0 ...................... (i)

According to the problem, α and β are the roots of the equation (i)

Therefore,

(i) Now α² + β² = (α + β)² - 2αβ = (4)² - 2 x 2 = 16 - 4 = 12.

(ii) α² - β² = (α + β)( α - β)

Now (α - β)2 = (α + β)² - 4αβ = (4)² - 4 x 2 = 16 - 8 = 8

⇒ α - β = ± √8

⇒ α - β = ± 2√2

Therefore, α² - β² = (α + β)( α - β) = 4 x (± 2√2) = ± 8√2.

(iii) α³ + β³ = (α + β)³ - 3αβ(α + β) = (4)³ - 3 x 2 x 4 = 64 - 24 = 40.

(iv)