Solve Quadratic Equations by Factoring
This is a tutorial on how to solve quadratic equations by factoring. There is also Factor Quadratic Expressions  Step by Step Calculator
Example 1 : Solve the following quadratic equation.
x ^{2}  3x = 0
Solution to Example 1:

Given
x ^{2}  3x = 0

Factor x out in the expression on the left.
x (x  3) = 0

For the product x (x  3) to be equal to zero we nedd to have
x = 0 or x  3 = 0

Solve the above simple equations to obtain the solutions.
x = 0
or
x = 3

As an exercise, check that x = 0 and x = 3 are solutions to the given equation.
Example 2 : Solve the quadratic equation given below
x ^{2}  5 x + 6 = 0
Solution to Example 2:

To factor the expression on the left, we need to write x ^{2}  5 x + 6 in the form factored:
x ^{2}  5 x + 6 = (x + a)(x + b)

so that the sum of a and b is 5 and their product is 6. The numbers that satisfy these conditions are  2 and  3. Hence
x ^{2}  5 x + 6 = (x  2)(x  3)

Substitute into the original equation and solve.
(x  2)(x  3) = 0

(x  2)(x  3) is equal to zero if
x  2 = 0
or
x  3 = 0

Solve the above equations to obtain two solutions to the given equation.
x = 2
or
x = 3

As an exercise, check that x = 0 and x = 3 are solutions to the given equation.
Example 3: Solve the following equation
2 x ^{2} + x  21 = 0
Solution to Example 3:

We first try to write 2 x ^{2} + x  21 in the factored form
2 x ^{2} + x  21 = (2x + a)(x + b)

Such that the product a b is equat to  21 and a + 2 b = 1
two pairs of numbers gives a product of  21: either 3 and 7 or 3 and 7. After some trial exercises it found that 2 x ^{2} + x  21 may be factored as follows:
2 x ^{2} + x  21 = (2x + 7)(x  3)

We now substitute into the original equation
(2x + 7)(x  3) = 0

and solve the following simpler equations
2x + 7 = 0
x  3 = 0

to obtain
x =  7 / 2
or x = 3

As an exercise, check that x = 0 and x = 3 are solutions to the given equation.
Example 4: Solve the following equation
(x  1)(x + 1 / 2) =  x + 1
Solution to Example 4:

At first we might be tempted into expanding the left side of the equation. However after examination of the right side, the above equation may be written as:
(x  1)(x + 1 / 2) =  (x  1)

Write the equation with the right side equal to zero.
(x  1)(x + 1 / 2) + (x  1) = 0

We now factor (x  1) out.
(x  1)(x + 1 / 2 + 1) = 0

and solve the following simpler equations
x  1 = 0
x + 3 / 2 = 0

to obtain
x = 1
or
x =  3 / 2
More references and links to quadratic equations.

