We know that, in mathematics calculations and solving problems are made easier by using formulae. In the same way, quadratic equation can be easily solved by using a formula. The quadratic formula, which is very useful for finding its roots can be derived using the method of completing the square. Let us derive the quadratic formula and learn how to use it for finding roots of the quadratic equations.
(c) Solution of a quadratic equation by formula method:
Consider the quadratic equation ax2 + bx + c = 0, a≠0
Divide the equation by a (i.e., coefficient of x2)
x2 + b/a x + c/a = 0
Find half the coefficient of x and square it,
1/2 x b/a = b/2a
(b/2a)2 = b^2/(4a)^2
Transpose the constant c/a to RHS
x2 + b/a x = – c/a
Add (b/2a)2 to both sides of the equation
x2 + b/a x + (b/2a)2 =(b/2a)2 – c/a
Factorise LHS and simplify RHS
(x + b/2a)2 = b^2/4a^2 – c/2a
Take square root on both sides of the equation
x + b/2a = ± √(b^2 – 4ac/4a^2) = ±√(b^2 – 4ac)/2a
x = –b/2a ±√(b^2 – 4ac)/2a
x = -b±√(b^2 – 4ac)/2a
Therefore, the roots of the quadratic equation ax2 + bx + x = 0 are –b – √(b^2 – 4ac)/2a and –b + √(b^2 – 4ac)/2a
x = -b±√(b^2 – 4ac)/2a is known as quadratic formula.
Example: Solve the quadratic equation x2 – 7x + 12 = 0 by formula method
Solution:
Given x2 – 7x + 12 = 0
The given quadratic equation x2 – 7x + 12 = 0 is of the form ax2 + bx + c = 0 where a = 1, b = – 7 and c = 12
Quadratic Equation – Exercise 9.5 – Class X
Solve the following quadratic equations by using the formula method.
- x2 – 4x + 2 = 0
- x2 – 2x + 4 = 0
- 2y2 + 6y = 3
- 15m2 – 11m + 2 = 0
- 8r2 = r + 2
- (2x + 3)(3x – 2) + 2 = 0
- a(x2 + 1) = x(a2 + 1)
- x2 + 8x + 6 = 0
Quadratic Equation – Exercise 9.5 – Solution:
Solve the following quadratic equations by using the formula method.
- x2 – 4x + 2 = 0
Solution:
x2 – 4x + 2 = 0
- x2 – 2x + 4 = 0
Solution:
x2 – 2x + 4 = 0
- 2y2 + 6y = 3
Solution:
2y2 + 6y – 3 = 0
- 15m2 – 11m + 2 = 0
Solution:
15m2 – 11m + 2 = 0
- 8r2 = r + 2
Solution:
8r2 – r – 2 = 0
- (2x + 3)(3x – 2) + 2 = 0
Solution:
(2x + 3)(3x – 2) + 2 = 0
6x2 – 4x + 9x – 6 + 2 = 0
6x2 + 5x – 4 = 0
- a(x2 + 1) = x(a2 + 1)
Solution:
a(x2 + 1) = x(a2 + 1)
ax2 + a = a2x + x
ax2 – a2x – x + a = 0
ax2 –x(a2 + 1) + a = 0
- x2 + 8x + 6 = 0
Solution:
x2 + 8x + 6 = 0
Quadratic Equations – Exercise 9.1 – Class X
Quadratic Equations – Exercise 9.2 – Class X
Quadratic equations – Exercise 9.3 – Class X
Quadratic Equations – Exercise 9.4 – Class X
