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# 10.3: Solving Quadratic Equations by Factoring

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## Factoring Method

To solve quadratic equations by factoring, we must make use of the zero-factor property.

##### Factoring Method
1. Set the equation equal to zero, that is, get all the nonzero terms on one side of the equal sign and 0 on the other.

$$ax^2 + bx + c = 0$$
2. Factor the quadratic expression.

$$()() = 0$$
3. By the zero-factor property, at least one of the factors must be zero, so, set each of the factors equal to 0 and solve for the variable.

## Sample Set A

Solve the following quadratic equations. (We will show the check for problem 1.)

##### Example $$\PageIndex{1}$$

$$\begin{array}{flushleft} x^2 - 7x + 12 &= 0 & & & \text{The equation is already set equal to } 0\\ (x-3)(x-4) &= 0 & & & \text{Factor. Set each factor equal to } 0.\\ x-3 &= 0 & \text{ or } x - 4 &= 0\\ x &= 3 & \text{ or } x &= 4 \end{array}$$

Check:

If $$x = 3, x^2 - 7x + 12 = 0$$

$$\begin{array}{flushleft} 3^2 - 7 \cdot 3 + 12 & = 0 & \text{Is this correct?}\\ 9 - 21 + 12 &= 0 & \text{Is this correct?}\\ 0 &= 0 & \text{Yes, this is correct?} \end{array}$$

Check: If $$x = 4, x^2 - 7x + 12 = 0$$

$$\begin{array}{flushleft} 4^2 - 7 \cdot 4 + 12 &= 0 & \text{Is this correct?}\\ 16 - 28 + 12 &= 0 & \text{Is this correct?}\\ 0 &= 0 & \text{Yes, this is correct} \end{array}$$

Thus, the solutions to this equation are $$x = 3, 4$$.

##### Example $$\PageIndex{2}$$

$$\begin{array}{flushleft} x^2 &= 25 & \text{Set the equation equal to } 0\\ x^2 - 25 &= 0 & \text{Factor.}\\ (x+5)(x-5) &= 0 & \text{Set each factor equal to } 0\\ x+5=0 & \text{ or } & x - 5 = 0\\ x = -5 & \text{ or } & x=5\\ \end{array}$$

Thus, the solutions to this equation are $$x = 5, -5$$.

##### Example $$\PageIndex{3}$$

$$\begin{array}{flushleft} x^2 &= 2x & \text{Set the equation equal to } 0\\ x^2 - 2x &= 0 & \text{Factor.}\\ x(x-2) && \text{Set each factor equal to} 0\\ x=0 & \text{ or } & x-2=0\\ && x=2 \end{array}$$

Thus, the solutions to this equation are $$x = 0, 2$$

##### Example $$\PageIndex{4}$$

$$\begin{array}{flushleft} 2x^2 + 7x - 15 &= 0 & \text{Factor.}\\ (2x - 3)(x+5) &= 0 & \text{Set each factor equal to } 0\\ 2x-3=0 & \text{ or } & x + 5 = 0\\ 2x=3 & \text{ or } & x=-5\\ x=\dfrac{3}{2} \end{array}$$

Thus, the solutions to this equation are $$x = \dfrac{3}{2}, -5$$.

##### Example $$\PageIndex{5}$$

$$63x^2 = 13x + 6$$

$$\begin{array}{flushleft} 63x^2 - 13x - 6 &= 0\\ (9x + 2)(7x - 3) &= 0\\ 9x + 2 = 0 & \text{ or } & 7x - 3 = 0\\ 9x = -2 & \text{ or } & 7x = 3\\ x=\dfrac{-2}{9} & \text{ or } & x = \dfrac{3}{7} \end{array}$$

Thus, the solutions to this equiation are $$x = \dfrac{-2}{9}, \dfrac{3}{7}$$

## Practice Set A

Solve the following equations, if possible.

##### Practice Problem $$\PageIndex{1}$$

$$(x−7)(x+4)=0$$

Answer

$$x=7, −4$$

##### Practice Problem $$\PageIndex{2}$$

$$(2x+5)(5x−7)=0$$

Answer

$$x = \dfrac{-5}{2}, \dfrac{7}{5}$$

##### Practice Problem $$\PageIndex{3}$$

$$x^2 + 2x - 24 = 0$$

Answer

$$x=4, −6$$

##### Practice Problem $$\PageIndex{4}$$

$$6x^2 + 13x - 5 = 0$$

Answer

$$x = \dfrac{1}{3}, \dfrac{-5}{2}$$

##### Practice Problem $$\PageIndex{5}$$

$$5y^2 + 2y = 3$$

Answer

$$y = \dfrac{3}{5}, -1$$

##### Practice Problem $$\PageIndex{6}$$

$$m(2m - 11) = 0$$

Answer

$$m = 0, \dfrac{11}{2}$$

##### Practice Problem $$\PageIndex{7}$$

$$6p^2 = -(5p + 1)$$

Answer

$$p = \dfrac{-1}{3}, \dfrac{-1}{2}$$

##### Practice Problem $$\PageIndex{8}$$

$$r^2 - 49 = 0$$

Answer

$$r=7,−7$$

## Solving Mentally After Factoring

Let's consider problems 4 and 5 of Sample Set A in more detail. Let's look particularly at the factorizations $$(2x-3)(x + 5) = 0$$ and $$(9x + 2)(7x - 3) = 0$$/ The next step is to set each factor equal to zero and solve. We can solve mentally if we understand how to solve linear equations: we transpose the constant from the variable term and then divide by the coefficient of the variable.

## Sample Set B

##### Example $$\PageIndex{6}$$

Solve the following equation mentally.

$$(2x - 3)(x + 5) = 0$$

$$\begin{array}{flushleft} 2x - 3 &= 0 & \text{Mentally add } 3 \text{ to both sides. The constant changes sign.}\\ 2x &= 3 & \text{Divide by } 2 \text{ the coefficient of } x \text{. The } 2 \text{ divides the cosntant } 3 \text{ into } \dfrac{3}{2}\\ & & \text{The coefficient becomes the denominator.}\\ x &= \dfrac{3}{2}\\ x + 5 &= 0 & \text{ Mentally subtract } 5 \text{ from both sides. The constant changes sign.}\\ x &= -5 & \text{Divide by the coefficient of } x, 1. \text{ The coefficient becomes the denominator}\\ x = \dfrac{-5}{1} &= -5\\ x &= -5 \end{array}$$

Now, we can immediately write the solution to the equation after factoring by looking at each factor, changing the sign of the constant, then divide by the coefficient.

## Practice Set B

##### Practice Problem $$\PageIndex{9}$$

Solve $$(9x + 2)(7x - 3) = 0$$ using this mental method.

Answer

$$x = -\dfrac{2}{9}, \dfrac{3}{7}$$

## Exercises

For the following problems, solve the equations, if possible.

##### Exercise $$\PageIndex{1}$$

$$(x+1)(x+3)=0$$

Answer

$$x=−1, −3$$

##### Exercise $$\PageIndex{2}$$

$$(x+4)(x+9)=0$$

##### Exercise $$\PageIndex{3}$$

$$(x−5)(x−1)=0$$

Answer

$$x=1, 5$$

##### Exercise $$\PageIndex{4}$$

$$(x−6)(x−3)=0$$

##### Exercise $$\PageIndex{5}$$

$$(x−4)(x+2)=0$$

Answer

$$x=−2, 4$$

##### Exercise $$\PageIndex{6}$$

$$(x+6)(x−1)=0$$

##### Exercise $$\PageIndex{7}$$

$$(2x+1)(x−7)=0$$

Answer

$$x = -\dfrac{1}{2}, 7$$

##### Exercise $$\PageIndex{8}$$

$$(3x+2)(x−1)=0$$

##### Exercise $$\PageIndex{9}$$

$$(4x+3)(3x−2)=0$$

Answer

$$x = -\dfrac{3}{4}, \dfrac{2}{3}$$

##### Exercise $$\PageIndex{10}$$

$$(5x−1)(4x+7)=0$$

##### Exercise $$\PageIndex{11}$$

$$(6x+5)(9x−4)=0$$

Answer

$$x = -\dfrac{5}{6}, \dfrac{4}{9}$$

##### Exercise $$\PageIndex{12}$$

$$(3a+1)(3a−1)=0$$

##### Exercise $$\PageIndex{13}$$

$$x(x+4)=0$$

Answer

$$x=−4, 0$$

##### Exercise $$\PageIndex{14}$$

$$y(y−5)=0$$

##### Exercise $$\PageIndex{15}$$

$$y(3y−4)=0$$

Answer

$$y = 0, \dfrac{4}{3}$$

##### Exercise $$\PageIndex{16}$$

$$b(4b+5)=0$$

##### Exercise $$\PageIndex{17}$$

$$x(2x+1)(2x+8)=0$$

Answer

$$x = -4, -\dfrac{1}{2}, 0$$

##### Exercise $$\PageIndex{18}$$

$$y(5y+2)(2y−1)=0$$

##### Exercise $$\PageIndex{19}$$

$$(x-8)^2 = 0$$

Answer

$$x=8$$

##### Exercise $$\PageIndex{20}$$

$$(x-2)^2 = 0$$

##### Exercise $$\PageIndex{21}$$

$$(b + 7)^2 = 0$$

Answer

$$b=−7$$

##### Exercise $$\PageIndex{22}$$

$$(a + 1)^2$$

##### Exercise $$\PageIndex{23}$$

$$(x(x-4)^2 = 0$$

Answer

$$x=0, 4$$

##### Exercise $$\PageIndex{24}$$

$$y(y + 9)^2 = 0$$

##### Exercise $$\PageIndex{25}$$

$$y(y-7)^2 = 0$$

Answer

$$y=0, 7$$

##### Exercise $$\PageIndex{26}$$

$$y(y + 5)^2 = 0$$

##### Exercise $$\PageIndex{27}$$

$$x^2 - 4 = 0$$

Answer

$$x=−2, 2$$

##### Exercise $$\PageIndex{28}$$

$$x^2 + 9 = 0$$

##### Exercise $$\PageIndex{29}$$

$$x^2 + 36$$

Answer

no solution

##### Exercise $$\PageIndex{30}$$

$$x^2 - 25 = 0$$

##### Exercise $$\PageIndex{31}$$

$$a^2 - 100 = 0$$

Answer

$$a=−10, 10$$

##### Exercise $$\PageIndex{32}$$

$$a^2 - 81 = 0$$

##### Exercise $$\PageIndex{33}$$

$$b^2 - 49 = 0$$

Answer

$$b=7, −7$$

##### Exercise $$\PageIndex{34}$$

$$y^2 - 1 = 0$$

##### Exercise $$\PageIndex{35}$$

$$3a^2 - 75 = 0$$

Answer

$$a=5, −5$$

##### Exercise $$\PageIndex{36}$$

$$5b^2 - 20 = 0$$

##### Exercise $$\PageIndex{37}$$

$$y^3 - y = 0$$

Answer

$$y=0, 1, −1$$

##### Exercise $$\PageIndex{38}$$

$$a^2 = 9$$

##### Exercise $$\PageIndex{39}$$

$$b^2 = 4$$

Answer

$$b=2, −2$$

##### Exercise $$\PageIndex{40}$$

$$b^2 = 1$$

##### Exercise $$\PageIndex{41}$$

$$a^2 = 36$$

Answer

$$a=6, −6$$

##### Exercise $$\PageIndex{42}$$

$$3a^2 = 12$$

##### Exercise $$\PageIndex{43}$$

$$-2x^2 = -4$$

Answer

$$x = \sqrt{2}, -\sqrt{2}$$

##### Exercise $$\PageIndex{44}$$

$$-2a^2 = -50$$

##### Exercise $$\PageIndex{45}$$

$$-7b^2 = -63$$

Answer

$$b=3, −3$$

##### Exercise $$\PageIndex{46}$$

$$-2x^2 = -32$$

##### Exercise $$\PageIndex{47}$$

$$3b^2 = 48$$

Answer

$$b=4, −4$$

##### Exercise $$\PageIndex{48}$$

$$a^2 - 8a + 16 = 0$$

##### Exercise $$\PageIndex{49}$$

$$y^2 + 10y + 25 = 0$$

Answer

$$y=−5$$

##### Exercise $$\PageIndex{50}$$

$$y^2 + 9y + 16 = 0$$

##### Exercise $$\PageIndex{51}$$

$$x^2 - 2x - 1 = 0$$

Answer

no solution

##### Exercise $$\PageIndex{52}$$

$$a^2 + 6a + 9 = 0$$

##### Exercise $$\PageIndex{53}$$

$$a^2 + 4a + 4 = 0$$

Answer

$$a=−2$$

##### Exercise $$\PageIndex{54}$$

$$x^2 + 12x = -36$$

##### Exercise $$\PageIndex{55}$$

$$b^2 - 14b = -49$$

Answer

$$b=7$$

##### Exercise $$\PageIndex{56}$$

$$3a^2 + 18a + 27 = 0$$

##### Exercise $$\PageIndex{57}$$

$$2m^3 + 4m^2 + 2m = 0$$

Answer

$$m=0, −1$$

##### Exercise $$\PageIndex{58}$$

$$3mn^2 - 36mn + 36m = 0$$

##### Exercise $$\PageIndex{59}$$

$$a^2 + 2a - 3 = 0$$

Answer

$$a=−3, 1$$

##### Exercise $$\PageIndex{60}$$

$$a^2 + 3a - 10 = 0$$

##### Exercise $$\PageIndex{61}$$

$$x^2 + 9x + 14 = 0$$

Answer

$$x=−7, −2$$

##### Exercise $$\PageIndex{62}$$

$$x^2 - 7x + 12 = 3$$

##### Exercise $$\PageIndex{63}$$

$$b^2 + 12b + 27 = 0$$

Answer

$$b=−9, −3$$

##### Exercise $$\PageIndex{64}$$

$$b^2 - 3b + 2 = 0$$

##### Exercise $$\PageIndex{65}$$

$$x^2 - 13x = -42$$

Answer

$$x=6, 7$$

##### Exercise $$\PageIndex{66}$$

$$a^3 = -8a^2 - 15a$$

##### Exercise $$\PageIndex{67}$$

$$6a^2 + 13a + 5 = 0$$

Answer

$$a = -\dfrac{5}{3}, -\dfrac{1}{2}$$

##### Exercise $$\PageIndex{68}$$

$$6x^2 - 4x - 2 = 0$$

##### Exercise $$\PageIndex{69}$$

$$12a^2 + 15a + 3 = 0$$

Answer

$$a = -\dfrac{1}{4}, -1$$

##### Exercise $$\PageIndex{70}$$

$$18b^2 + 24b + 6 = 0$$

##### Exercise $$\PageIndex{71}$$

$$12a^2 + 24a + 12 = 0$$

Answer

$$a=−1$$

##### Exercise $$\PageIndex{72}$$

$$4x^2 - 4x = -1$$

##### Exercise $$\PageIndex{73}$$

$$2x^2 = x + 15$$

Answer

$$x = -\dfrac{5}{2}, 3$$

##### Exercise $$\PageIndex{74}$$

$$4a^2 = 4a + 3$$

##### Exercise $$\PageIndex{75}$$

$$4y^2 = -4y - 2$$

Answer

no solution

##### Exercise $$\PageIndex{76}$$

$$9y^2 = 9y + 18$$

## Exercises For Review

##### Exercise $$\PageIndex{77}$$

Simplify $$(x^4y^3)^2(xy^2)^4$$

Answer

$$x^{12}y^{14}$$

##### Exercise $$\PageIndex{78}$$

Write $$(x^{-2}y^3w^4)^{-2}$$ so that only positive exponents appear.

##### Exercise $$\PageIndex{79}$$

Find the sum: $$\dfrac{x}{x^2 - x - 2} + \dfrac{1}{x^2 - 3x + 2}$$

Answer

$$\dfrac{x^2 + 1}{(x+1)(x-1)(x-2)}$$

##### Exercise $$\PageIndex{80}$$

Simplify $$\dfrac{\dfrac{1}{a} + \dfrac{1}{b}}{\dfrac{1}{a} - \dfrac{1}{b}}$$

##### Exercise $$\PageIndex{81}$$

Solve $$(x + 4)(3x + 1) = 0$$

Answer

$$x = -4, \dfrac{-1}{3}$$

This page titled 10.3: Solving Quadratic Equations by Factoring is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Denny Burzynski & Wade Ellis, Jr. (OpenStax CNX) .

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