Solving Quadratic Equations using Factoring - Examples, Exercises and Solutions

Let's first expand the parentheses using the formula:

$a(x+b)=ax+ab$

$(2\times x)+(2\times4)+8=0$

$2x+8+8=0$

Next, we will substitute in our terms accordingly:

$2x+16=0$

Then, we will move the 16 to the left-hand side, keeping the appropriate sign:

$2x=-16$

Finally, we divide both sides by 2:

$\frac{2x}{2}=-\frac{16}{2}$

$x=-8$

To open parentheses we will use the formula:

$a(x+b)=ax+ab$

$(7\times-2x)+(7\times5)=77$

We multiply accordingly

$-14x+35=77$

We will move the 35 to the right section and change the sign accordingly:

$-14x=77-35$

We solve the subtraction exercise on the right side and we will obtain:

$-14x=42$

We divide both sections by -14

$\frac{-14x}{-14}=\frac{42}{-14}$

$x=-3$

We open the parentheses according to the formula:

$a(x+b)=ax+ab$

$5\times x+5\times3=0$

$5x+15=0$

We will move the 15 to the right section and keep the corresponding sign:

$5x=-15$

Divide both sections by 5

$\frac{5x}{5}=\frac{-15}{5}$

$x=-3$

We open the parentheses on the left side by the distributive property and use the formula:

$a(x+b)=ax+ab$

$-\frac{3}{2}x-12=\frac{1}{2}$

We multiply all terms by 2 to get rid of the fractions:

$-3x-12\times2=1$

$-3x-24=1$

We will move the minus 24 to the right section and keep the corresponding sign:

$-3x=24+1$

$-3x=25$

Divide both sections by minus 3:

$\frac{-3x}{-3}=\frac{25}{-3}$

$x=-\frac{25}{3}$

To open the parentheses on the left side, we'll use the formula:

$-a\left(b+c\right)=-ab-ac$

$-12a-24=27a$

We'll arrange the equation so that the terms with 'a' are on the right side, and maintain the plus and minus signs during the transfer:

$-24=27a+12a$

Let's group the terms on the right side:

$-24=39a$

Let's divide both sides by 39:

$-\frac{24}{39}=\frac{39a}{39}$

$-\frac{24}{39}=a$

Note that we can reduce the fraction since both numerator and denominator are divisible by 3:

$-\frac{8}{13}=a$

Let's open the parentheses using the distributive property and use the formula:

$a(x+b)=ax+ab$

$40+5a-2a-14=56$

We'll substitute the terms accordingly:

$26+3a=56$

We'll move 26 to the right side and keep the appropriate sign:

$3a=56-26$

$3a=30$

We'll divide both sides by 3:

$\frac{3a}{3}=\frac{30}{3}$

$a=10$

We will use the extended division rule and the formula:

$a(x+b)=ax+ab$

$-42x+36+5+8x=0$

Let's input the appropriate terms:

$-34x+41=0$

We'll move -34x to the right side and maintain the appropriate sign:

$41=34x$

Let's divide both sides by 34:

$\frac{41}{34}=\frac{34x}{34}$

$\frac{41}{34}=x$

We'll convert the simple fraction to a mixed fraction:

$x=1\frac{7}{34}$

To open parentheses we will use the formula:

$a\left(x+b\right)=ax+ab$

$(-8\times2x)+(-8\times4)=(6\times x)+(6\times-4)+3$

We multiply accordingly:

$-16x-32=6x-24+3$

Calculate the elements on the right section:

$-16x-32=6x-21$

In the left section we enter the elements with the X and in the left section those without the X, remember to change the plus and minus signs as appropriate when transferring:

$-32+21=6x+16x$

Calculate the elements accordingly

$-11=22x$

We divide the two sections by 22

$-\frac{11}{22}=\frac{22x}{22}$

$-\frac{1}{2}=x$

To open parentheses we will use the formula:

$a(x+b)=ax+ab$

$(8\times x)+(8\times-2)=(-4\times x)+(-4\times3)$

We multiply accordingly

$8x-16=-4x-12$

In the left section we enter the elements with the X and in the right section those without the X, remember to change the plus and minus signs as appropriate when transferring:

$8x+4x=-12+16$

We solve accordingly

$12x=4$

We divide both sections by 12

$\frac{12x}{12}=\frac{4}{12}$

We reduce and obtain$x=\frac{4}{12}=\frac{1}{3}$

We open the parentheses in both sections by the distributive property and use the formula:

$a(x+b)=ax+ab$

$-18+9x=3x+12$

We move 3X to the left section, and 18 to the right section and maintain the corresponding signs:

$9x-3x=12+18$

We add the terms:

$6x=30$

We divide both sections by 6:

$\frac{6x}{6}=\frac{30}{6}$

$x=5$

To open parentheses we will use the formula:

$a\left(x+b\right)=ax+ab$

$(-7\times2x)+(-7\times3)+(-4\times x)+(-4\times2)=(5\times2)+(5\times-3x)$

We multiply accordingly:

$-14x-21-4x-8=10-15x$

We calculate the elements in the left section:

$-18x-29=10-15x$

In the left section we enter the elements with the X and in the right section those without the X, remember to change the plus and minus signs as appropriate when transferring:

$-18x+15x=10+29$

We calculate the elements accordingly:

$-3x=39$

We divide the two sections by -3:

$\frac{-3x}{-3}=\frac{39}{-3}$

$x=-13$