Solve (5x+8)^2=36

$$( 5 x + 8 ) ^ { 2 } = 36$$

Solve for x

$$x=-\frac{2}{5}=-0.4$$

$$x = -\frac{14}{5} = -2\frac{4}{5} = -2.8$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(5x+8\right)^{2}.

$$25x^{2}+80x+64=36$$

Subtract 36 from both sides.

$$25x^{2}+80x+64-36=0$$

Subtract 36 from 64 to get 28.

$$25x^{2}+80x+28=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 25x^{2}+ax+bx+28. To find a and b, set up a system to be solved.

$$a+b=80 ab=25\times 28=700$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 700.

$$1,700 2,350 4,175 5,140 7,100 10,70 14,50 20,35 25,28$$

Calculate the sum for each pair.

$$1+700=701 2+350=352 4+175=179 5+140=145 7+100=107 10+70=80 14+50=64 20+35=55 25+28=53$$

The solution is the pair that gives sum 80.

$$a=10 b=70$$

Rewrite 25x^{2}+80x+28 as \left(25x^{2}+10x\right)+\left(70x+28\right).

$$\left(25x^{2}+10x\right)+\left(70x+28\right)$$

Factor out 5x in the first and 14 in the second group.

$$5x\left(5x+2\right)+14\left(5x+2\right)$$

Factor out common term 5x+2 by using distributive property.

$$\left(5x+2\right)\left(5x+14\right)$$

To find equation solutions, solve 5x+2=0 and 5x+14=0.

$$x=-\frac{2}{5} x=-\frac{14}{5}$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(5x+8\right)^{2}.

$$25x^{2}+80x+64=36$$

Subtract 36 from both sides.

$$25x^{2}+80x+64-36=0$$

Subtract 36 from 64 to get 28.

$$25x^{2}+80x+28=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 25 for a, 80 for b, and 28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-80±\sqrt{80^{2}-4\times 25\times 28}}{2\times 25}$$

Square 80.

$$x=\frac{-80±\sqrt{6400-4\times 25\times 28}}{2\times 25}$$

Multiply -4 times 25.

$$x=\frac{-80±\sqrt{6400-100\times 28}}{2\times 25}$$

Multiply -100 times 28.

$$x=\frac{-80±\sqrt{6400-2800}}{2\times 25}$$

Add 6400 to -2800.

$$x=\frac{-80±\sqrt{3600}}{2\times 25}$$

Take the square root of 3600.

$$x=\frac{-80±60}{2\times 25}$$

Multiply 2 times 25.

$$x=\frac{-80±60}{50}$$

Now solve the equation x=\frac{-80±60}{50} when ± is plus. Add -80 to 60.

$$x=-\frac{20}{50}$$

Reduce the fraction \frac{-20}{50} to lowest terms by extracting and canceling out 10.

$$x=-\frac{2}{5}$$

Now solve the equation x=\frac{-80±60}{50} when ± is minus. Subtract 60 from -80.

$$x=-\frac{140}{50}$$

Reduce the fraction \frac{-140}{50} to lowest terms by extracting and canceling out 10.

$$x=-\frac{14}{5}$$

The equation is now solved.

$$x=-\frac{2}{5} x=-\frac{14}{5}$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(5x+8\right)^{2}.

$$25x^{2}+80x+64=36$$

Subtract 64 from both sides.

$$25x^{2}+80x=36-64$$

Subtract 64 from 36 to get -28.

$$25x^{2}+80x=-28$$

Divide both sides by 25.

$$\frac{25x^{2}+80x}{25}=-\frac{28}{25}$$

Dividing by 25 undoes the multiplication by 25.

$$x^{2}+\frac{80}{25}x=-\frac{28}{25}$$

Reduce the fraction \frac{80}{25} to lowest terms by extracting and canceling out 5.

$$x^{2}+\frac{16}{5}x=-\frac{28}{25}$$

Divide \frac{16}{5}, the coefficient of the x term, by 2 to get \frac{8}{5}. Then add the square of \frac{8}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{16}{5}x+\left(\frac{8}{5}\right)^{2}=-\frac{28}{25}+\left(\frac{8}{5}\right)^{2}$$

Square \frac{8}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{16}{5}x+\frac{64}{25}=\frac{-28+64}{25}$$

Add -\frac{28}{25} to \frac{64}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{16}{5}x+\frac{64}{25}=\frac{36}{25}$$

Factor x^{2}+\frac{16}{5}x+\frac{64}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{8}{5}\right)^{2}=\frac{36}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{8}{5}\right)^{2}}=\sqrt{\frac{36}{25}}$$

Simplify.

$$x+\frac{8}{5}=\frac{6}{5} x+\frac{8}{5}=-\frac{6}{5}$$

Subtract \frac{8}{5} from both sides of the equation.

$$x=-\frac{2}{5} x=-\frac{14}{5}$$