Solve (5+8x)^2=(5+x)(5+29x)

$$( 5 + 8 x ) ^ { 2 } = ( 5 + x ) ( 5 + 29 x )$$

Solve for x

$$x=2$$

$$x=0$$

Steps Using Factoring

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

$$25+80x+64x^{2}=\left(5+x\right)\left(5+29x\right)$$

Use the distributive property to multiply 5+x by 5+29x and combine like terms.

$$25+80x+64x^{2}=25+150x+29x^{2}$$

Subtract 25 from both sides.

$$25+80x+64x^{2}-25=150x+29x^{2}$$

Subtract 25 from 25 to get 0.

$$80x+64x^{2}=150x+29x^{2}$$

Subtract 150x from both sides.

$$80x+64x^{2}-150x=29x^{2}$$

Combine 80x and -150x to get -70x.

$$-70x+64x^{2}=29x^{2}$$

Subtract 29x^{2} from both sides.

$$-70x+64x^{2}-29x^{2}=0$$

Combine 64x^{2} and -29x^{2} to get 35x^{2}.

$$-70x+35x^{2}=0$$

Factor out x.

$$x\left(-70+35x\right)=0$$

To find equation solutions, solve x=0 and -70+35x=0.

$$x=0 x=2$$

Steps Using the Quadratic Formula

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

$$25+80x+64x^{2}=\left(5+x\right)\left(5+29x\right)$$

Use the distributive property to multiply 5+x by 5+29x and combine like terms.

$$25+80x+64x^{2}=25+150x+29x^{2}$$

Subtract 25 from both sides.

$$25+80x+64x^{2}-25=150x+29x^{2}$$

Subtract 25 from 25 to get 0.

$$80x+64x^{2}=150x+29x^{2}$$

Subtract 150x from both sides.

$$80x+64x^{2}-150x=29x^{2}$$

Combine 80x and -150x to get -70x.

$$-70x+64x^{2}=29x^{2}$$

Subtract 29x^{2} from both sides.

$$-70x+64x^{2}-29x^{2}=0$$

Combine 64x^{2} and -29x^{2} to get 35x^{2}.

$$-70x+35x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$35x^{2}-70x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 35 for a, -70 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-70\right)±\sqrt{\left(-70\right)^{2}}}{2\times 35}$$

Take the square root of \left(-70\right)^{2}.

$$x=\frac{-\left(-70\right)±70}{2\times 35}$$

The opposite of -70 is 70.

$$x=\frac{70±70}{2\times 35}$$

Multiply 2 times 35.

$$x=\frac{70±70}{70}$$

Now solve the equation x=\frac{70±70}{70} when ± is plus. Add 70 to 70.

$$x=\frac{140}{70}$$

Divide 140 by 70.

$$x=2$$

Now solve the equation x=\frac{70±70}{70} when ± is minus. Subtract 70 from 70.

$$x=\frac{0}{70}$$

Divide 0 by 70.

$$x=0$$

The equation is now solved.

$$x=2 x=0$$

Steps for Completing the Square

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

$$25+80x+64x^{2}=\left(5+x\right)\left(5+29x\right)$$

Use the distributive property to multiply 5+x by 5+29x and combine like terms.

$$25+80x+64x^{2}=25+150x+29x^{2}$$

Subtract 150x from both sides.

$$25+80x+64x^{2}-150x=25+29x^{2}$$

Combine 80x and -150x to get -70x.

$$25-70x+64x^{2}=25+29x^{2}$$

Subtract 29x^{2} from both sides.

$$25-70x+64x^{2}-29x^{2}=25$$

Combine 64x^{2} and -29x^{2} to get 35x^{2}.

$$25-70x+35x^{2}=25$$

Subtract 25 from both sides.

$$-70x+35x^{2}=25-25$$

Subtract 25 from 25 to get 0.

$$-70x+35x^{2}=0$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$35x^{2}-70x=0$$

Divide both sides by 35.

$$\frac{35x^{2}-70x}{35}=\frac{0}{35}$$

Dividing by 35 undoes the multiplication by 35.

$$x^{2}+\left(-\frac{70}{35}\right)x=\frac{0}{35}$$

Divide -70 by 35.

$$x^{2}-2x=\frac{0}{35}$$

Divide 0 by 35.

$$x^{2}-2x=0$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=1$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{1}$$

Simplify.

$$x-1=1 x-1=-1$$

Add 1 to both sides of the equation.

$$x=2 x=0$$