Solve Quadratic Equations by Factoring

This is a tutorial questions on how to solve quadratic equations by factoring. The detailed solutions to thsese questions are included. There is also Factor Quadratic Expressions - Step by Step Calculator in this website.

Questions with Solutions

Solve the following quadratic equation.
x ² - 3x = 0

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.

Solve the quadratic equation given below
x ² - 5 x + 6 = 0

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.

Solve the following equation
2 x ² + x - 21 = 0

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 equal 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.

Solve the following equation
(x - 1)(x + 1 / 2) = - x + 1

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