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

Question 1

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.

Question 2

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.

Question 3

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.

Question 4

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

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