Solve left(2+4xright)^2=(2+x)(2+13x)

$${ \left(2+4x \right) }^{ 2 } =(2+x)(2+13x)$$

Solve for x

$$x=4$$

$$x=0$$

Steps Using Factoring

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2+4x\right)^{2}.

$$4+16x+16x^{2}=\left(2+x\right)\left(2+13x\right)$$

Use the distributive property to multiply 2+x by 2+13x and combine like terms.

$$4+16x+16x^{2}=4+28x+13x^{2}$$

Subtract 4 from both sides.

$$4+16x+16x^{2}-4=28x+13x^{2}$$

Subtract 4 from 4 to get 0.

$$16x+16x^{2}=28x+13x^{2}$$

Subtract 28x from both sides.

$$16x+16x^{2}-28x=13x^{2}$$

Combine 16x and -28x to get -12x.

$$-12x+16x^{2}=13x^{2}$$

Subtract 13x^{2} from both sides.

$$-12x+16x^{2}-13x^{2}=0$$

Combine 16x^{2} and -13x^{2} to get 3x^{2}.

$$-12x+3x^{2}=0$$

Factor out x.

$$x\left(-12+3x\right)=0$$

To find equation solutions, solve x=0 and -12+3x=0.

$$x=0 x=4$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2+4x\right)^{2}.

$$4+16x+16x^{2}=\left(2+x\right)\left(2+13x\right)$$

Use the distributive property to multiply 2+x by 2+13x and combine like terms.

$$4+16x+16x^{2}=4+28x+13x^{2}$$

Subtract 4 from both sides.

$$4+16x+16x^{2}-4=28x+13x^{2}$$

Subtract 4 from 4 to get 0.

$$16x+16x^{2}=28x+13x^{2}$$

Subtract 28x from both sides.

$$16x+16x^{2}-28x=13x^{2}$$

Combine 16x and -28x to get -12x.

$$-12x+16x^{2}=13x^{2}$$

Subtract 13x^{2} from both sides.

$$-12x+16x^{2}-13x^{2}=0$$

Combine 16x^{2} and -13x^{2} to get 3x^{2}.

$$-12x+3x^{2}=0$$

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.

$$3x^{2}-12x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -12 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}}}{2\times 3}$$

Take the square root of \left(-12\right)^{2}.

$$x=\frac{-\left(-12\right)±12}{2\times 3}$$

The opposite of -12 is 12.

$$x=\frac{12±12}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{12±12}{6}$$

Now solve the equation x=\frac{12±12}{6} when ± is plus. Add 12 to 12.

$$x=\frac{24}{6}$$

Divide 24 by 6.

$$x=4$$

Now solve the equation x=\frac{12±12}{6} when ± is minus. Subtract 12 from 12.

$$x=\frac{0}{6}$$

Divide 0 by 6.

$$x=0$$

The equation is now solved.

$$x=4 x=0$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2+4x\right)^{2}.

$$4+16x+16x^{2}=\left(2+x\right)\left(2+13x\right)$$

Use the distributive property to multiply 2+x by 2+13x and combine like terms.

$$4+16x+16x^{2}=4+28x+13x^{2}$$

Subtract 28x from both sides.

$$4+16x+16x^{2}-28x=4+13x^{2}$$

Combine 16x and -28x to get -12x.

$$4-12x+16x^{2}=4+13x^{2}$$

Subtract 13x^{2} from both sides.

$$4-12x+16x^{2}-13x^{2}=4$$

Combine 16x^{2} and -13x^{2} to get 3x^{2}.

$$4-12x+3x^{2}=4$$

Subtract 4 from both sides.

$$-12x+3x^{2}=4-4$$

Subtract 4 from 4 to get 0.

$$-12x+3x^{2}=0$$

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.

$$3x^{2}-12x=0$$

Divide both sides by 3.

$$\frac{3x^{2}-12x}{3}=\frac{0}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\left(-\frac{12}{3}\right)x=\frac{0}{3}$$

Divide -12 by 3.

$$x^{2}-4x=\frac{0}{3}$$

Divide 0 by 3.

$$x^{2}-4x=0$$

Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-4x+\left(-2\right)^{2}=\left(-2\right)^{2}$$

Square -2.

$$x^{2}-4x+4=4$$

Factor x^{2}-4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-2\right)^{2}=4$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{4}$$

Simplify.

$$x-2=2 x-2=-2$$

Add 2 to both sides of the equation.

$$x=4 x=0$$