Solving Quadratic Equations by Factoring
July 28, 2025
Introduction
A quadratic equation is a polynomial equation of the form:
where a, \, b,\, and c are real numbers and a\neq0.
There are some instances wherein factoring works best as technique in solving quadratic equations, before we proceed to applying the said technique, let us understand first this property:
For any real numbers a and b , if ab = 0, then a = 0 \text{ or } b = 0.
Basically, if we have two factors, and their product is 0, then we can equate both factors to 0 to solve for the value of the desired variable. Now, to apply this technique, here are the guidelines:
Follow these steps to solve a quadratic equation by factoring:
1. Ensure the equation is written as \,ax^2 + bx + c = 0\, .
2. Rewrite ax^2 + bx + c as a product of two binomials by applying factoring techniques.
3. Apply the Zero Product Property by setting each factor equal to zero.
4. Solve for x, and simplify, if necessary.
Illustrative Example #1
Find the solutions of the equation x^2 - 5x + 6 = 0 using factoring.
Solution:
Thus, the solutions of x^2 - 5x + 6 = 0 are x = 2 \text{ and } x = -3.
Illustrative Example #2
Find the solutions of the equation x^2 - 3x - 18 = 0 using factoring.
Solution:
Thus, the solutions of x^2 - 3x - 18 = 0 are x = 6 and x = -3.
Illustrative Example #3
Find the solutions of the equation 10x^2 - 3x - 4 = 0 using factoring.
Solution:
Thus, the solutions of 10x^2 - 3x - 4 = 0 are x = -\dfrac12 and x = \dfrac45.
Illustrative Example #4
Find the solutions of the equation 3x^2 - x - 10 = 0 using factoring.
Solution:
Thus, the solutions of 3x^2 - x - 10 = 0 are x = -\dfrac{5}{3} and x = 2.
Illustrative Example #5
Find the solutions of the equation x^2 - 144 = 0 using factoring.
Solution:
Thus, the solutions of x^2 - 144 = 0 are x = 12 and x = -12.
Illustrative Example #6
Find the solutions of the equation x^2 - 16x + 64 = 0 using factoring.
Solution:
Thus, the solution of x^2 - 16x + 64 = 0 is x = 8 (a repeated root).
Illustrative Example #7
Find the solutions of the equation x^2 - 12x + 9 = 0 using factoring.
Solution:
Thus, the solution of 4x^2 - 12x + 9 = 0 is x = \dfrac{3}{2} (a repeated root).
Illustrative Example #8
Find the solutions of the equation x^2 - 25 = 0 using factoring.
Solution:
Thus, the solution of x^2 - 25 = 0 are x = 5 and x = -5.
Illustrative Example #9
Find the solutions of the equation 2x^2 - 3x + 1 = 0 using factoring.
Solution:
Thus, the solution of 2x^2 - 3x + 1 = 0 are x = \dfrac{1}{2} and x = 1.
Illustrative Example #10
Find the solutions of the equation 12x^2 - 7x - 12 = 0 using factoring.
Solution:
Thus, the solution of 12x^2 - 7x - 12 = 0 are x = \dfrac{4}{3} and x = -\dfrac{3}{4}.
