New Arrivals

Solution of Quadratic Equations

To find the solution set of a quadratic equation, following methods are used:

  1. factorization
  2. completing square
  3. use of quadratic formula

Solution by factorization

In this method, write the quadratic equation in the standard form as

ax2 + bx + c = 0       (i)

If two numbers r and s can be found for the equation (i) such that

r + s = b       and       rs = ac

then ax2 + bx + c can be factorized into two linear factors.

Example

Solve the quadratic equation 3x2 − 6x = x + 20 by factorization.

 Solution:

3x2 − 6x = x + 20       (i)

The standard form of (i) is

3x2 − 7x − 20 = 0       (ii)

Here    a = 3, b = −7, c = −20

and      ac = 3 × (−20) = −60

As       −12 + 5 = −7 and −12 × 5 = −60, so

the equation (ii) can be written as

3x2 − 12x + 5x − 20 = 0

or        3x (x − 4) + 5 (x − 4) = 0

⇒        (x − 4) (3x + 5) = 0

Either  x − 4 = 0      or      3x + 5 = 0

x = 4      or      3x = −5     ⇒  x = −5⁄3

∴    x = −5⁄3, 4 are the solutions of the given equation.

Thus, the solution set is {−5⁄3, 4}.

Activity:

Factorize   x2 − x − 2 = 0.

 Solution:

x2 − x − 2 = 0       (i)

Here    a = 1, b = −1, c = −2

and      ac = 1 × (−2) = −2

As       −2 + 1 = −1 and −2 × 1 = −2, so

the equation (i) can be written as

x2 − 2x + x − 2 = 0

or        x (x − 2) + 1 (x − 2) = 0

⇒        (x − 2) (x + 1) = 0

∴    (x − 2), (x + 1) are factors of the given equation.

Example

Solve 5x2 = 30x by factorization.

 Solution:

5x2 = 30x

Remember that cancelling of x on both sides of 5x2 = 30x means the loss of one root, i.e., x = 0

5x2 − 30x = 0       (i)

The equation (i) is factorized as

5x (x − 6) = 0

Either  5x = 0   or   x − 6 = 0     ⇒   x = 0   or   x = 6

∴    x = 0, 6 are the roots of the given equation.

Thus, the solution set is {0, 6}.

Leave a Reply

Your email address will not be published. Required fields are marked *

*