To solve a quadratic equation by factoring:

  1. Put it in standard form: ax2+bx+c=0

  2. Factor the left-hand side

  3. Use the Zero Factor Law


Solve: x^{2}=2-x 

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

x^{2}=2-x Original equation
x^{2}+x-2=0 Put in standard form: subtract 2 from both sides; add x to both sides
(x+2)(x-1)=0 Factor the left-hand side
x+2=0 or x-1=0 Use the Zero Factor Law
x=-2 or x=1 Solve the simpler equations


Check by substituting into the original equation:

(-2)^{2}=2-(-2); 4=4; Check!

(1)^{2}=2-1; 1=1 Check!


Solve: (x+3)(x-2)=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.

(x+3)(x-2)=0 Original equation
x+3=0 or x-2=0 Use the Zero Factor Law
x=-3 or x=2 Solve the simpler equations


Check by substituting into the original equation:

(-3+3)(-3-2)=0;  0=0 Check!

(2+3)(2-2)=0; 0=0 Check!


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

Solution: Again, do not multiply it out! When you have a product on one side, and zero on the other side, then you are all set to use the Zero Factor Law.

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



Check by substituting into the original equation:

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

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


Solve: x^{2}+4x-5=0

Solution: Note that it is already in standard form.

x^{2}+4x-5=0 Original equation
(x+5)(x-1)=0 Factor the left-hand side
x+5=0 or x-1=0 Use the Zero Factor Law
x=-5 or x=1 Solve the simpler equations


Check by substituting into the original equation:

(-5)^{2}+4(-5)-5=0; 25-20-5=0; 0=0 Check!

1^{2}+4(1)-5+0; 1+4-5=0; 0=0 Check!


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

Solution:

14=-5x+x^{2} Original equation
x^{2}-5x-14=0 Put in standard form: subtract 14 from both sides; write in the conventional way
(x-7)(x+2)=0 Factor the left-hand side
x-7=0 or x+2=0 Use the Zero Factor Law
x=7 or x=-2 Solve the simpler equations


Check by substituting into the original equation:

14=-5(7)+7^{2}; 14=-35+49; 14=14 Check!

14=-5(-2)+(-2)^{2}; 14=10+4; 14=14 Check!


Solve: 6x=2x^{2}

Solution: When there is no constant term, the factoring is much easier.

6x=2x^{2} Original equation
2x^{2}-6x=0 Put in standard form: subtract 6x from both sides; write in the conventional way
x^{2}-3x=0 Optional step: divide both sides by 2
x(x-3)=0 Factor the left-hand side
x=0 or x-3=0 Use the Zero Factor Law
x=0 or x=3 Solve the simpler equations


Check by substituting into the original equation:

6\cdot 0=2\cdot 0^{2}; 0=0; Check!

6\cdot 3=2\cdot 3^{2}; 18=18; Check!


Source: Tree of Math, https://www.onemathematicalcat.org/algebra_book/online_problems/solve_quad_eq_simple_fac.htm
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Last modified: Wednesday, 5 May 2021, 2:31 PM