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.

Question1

Solve the following quadratic equation.

x 2 - 3x = 0

Solution to Question1

  • 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.

Question2

Solve the quadratic equation given below

x 2 - 5 x + 6 = 0

Solution to Question2

  • 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.

Question3

Solve the following equation

2 x 2 + x - 21 = 0

Solution to Question3

  • 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.

Question4

Solve the following equation

(x - 1)(x + 1 / 2) = - x + 1

Solution to Question4

  • 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

Quadratic Equations Calculator and Solver.
Solve Equations, Systems of Equations and Inequalities.