Using Factoring to Solve Complicated Quadratic Equations

Read each example slowly and try to identify the factoring methods being used and why each step is performed. After you have reviewed the materials, complete a few practice problems and check your answers.

To solve a quadratic equation by factoring:

  • Put it in standard form: ax^{2}+bx+c=0
  • Factor the left-hand side
  • Use the Zero Factor Law


Solve: 3x^{2}=5-14x

Solution: Write a nice, clean list of equivalent equations.

3x^{2}=5-14x Original equation
3x^{2}+14x-5=0 Put in standard form: subtract 5 from both sides; add 14x to both sides
(3x-1)(x+5)=0 Factor the left-hand side; you may want to use the factor by grouping method
3x-1=0 or x+5=0 Use the Zero Factor Law
3x=1 or x=-5 Solve the simpler equations
x=\frac{1}{3} or x=-5 Solve the simpler equations


Check by substituting into the original equation:

3(\frac{1}{3})^{2}=5-14(\frac{1}{3});  3\cdot \frac{1}{9}=\frac{15}{3}-\frac{14}{3};  \frac{1}{3}=\frac{1}{3};  Check!

3(-5)^{2}=5-14(-5); 3\cdot 25=5+70;  75=75  Check!


Solve: (2x+3)(5x-1)=0

Solution: Do not multiply it out!

If it is already in factored form, with zero on one side, then be happy that a lot of the work has already been done for you.

(2x+3)(5x-1)=0 Original equation
2x+3=0 or 5x-1=0 Use the Zero Factor Law
2x=-3 or 5x=1 Solve the simpler equations
x=-\frac{3}{2} or x=\frac{1}{5} Solve the simpler equations


Check by substituting into the original equation:

(2(-\frac{3}{2})+3)(5(-\frac{3}{2})-1)=0;  0=0;  Check!

(2(\frac{1}{5})+3)(5(\frac{1}{5})-1)=0;  0=0;  Check!


Solve: 10x^{2}-11x-6=0

Solution: Note that it is already in standard form.


10x^{2}-11x-6=0 Original equation
(5x+2)(2x-3)=0 Factor the left-hand side; you may want to use the factor by grouping method
5x+2=0 or 2x-3=0 Use the Zero Factor Law
5x=-2 or 2x=3 Solve the simpler equations
x=-\frac{2}{5} or x=\frac{3}{2} Solve the simpler equations


Check by substituting into the original equation:

10(-\frac{2}{3})^{2}-11(-\frac{2}{5})-6=0;  10(\frac{4}{25})+\frac{22}{5}-6=0;  2(\frac{4}{5})+\frac{22}{5}-\frac{30}{5}=0;  0=0;  Check!

10(\frac{3}{2})^{2}-11(\frac{3}{2})-6=0;  10(\frac{9}{4})-\frac{33}{2}-6=0;  5(\frac{9}{2})-\frac{33}{2}-\frac{12}{2}=0; 0=0;  Check!



Source: Tree of Math, https://www.onemathematicalcat.org/algebra_book/online_problems/solve_quad_eq_morecomp_fac.htm
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Last modified: Thursday, May 6, 2021, 10:11 AM