Solve (1+4x)^2=(1+x)(13x+1)

$$( 1 + 4 x ) ^ { 2 } = ( 1 + x ) ( 13 x + 1 )$$

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(1+4x\right)^{2}.

$$1+8x+16x^{2}=\left(1+x\right)\left(13x+1\right)$$

Use the distributive property to multiply 1+x by 13x+1 and combine like terms.

$$1+8x+16x^{2}=14x+1+13x^{2}$$

Subtract 14x from both sides.

$$1+8x+16x^{2}-14x=1+13x^{2}$$

Combine 8x and -14x to get -6x.

$$1-6x+16x^{2}=1+13x^{2}$$

Subtract 1 from both sides.

$$1-6x+16x^{2}-1=13x^{2}$$

Subtract 1 from 1 to get 0.

$$-6x+16x^{2}=13x^{2}$$

Subtract 13x^{2} from both sides.

$$-6x+16x^{2}-13x^{2}=0$$

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

$$-6x+3x^{2}=0$$

Factor out x.

$$x\left(-6+3x\right)=0$$

To find equation solutions, solve x=0 and -6+3x=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(1+4x\right)^{2}.

$$1+8x+16x^{2}=\left(1+x\right)\left(13x+1\right)$$

Use the distributive property to multiply 1+x by 13x+1 and combine like terms.

$$1+8x+16x^{2}=14x+1+13x^{2}$$

Subtract 14x from both sides.

$$1+8x+16x^{2}-14x=1+13x^{2}$$

Combine 8x and -14x to get -6x.

$$1-6x+16x^{2}=1+13x^{2}$$

Subtract 1 from both sides.

$$1-6x+16x^{2}-1=13x^{2}$$

Subtract 1 from 1 to get 0.

$$-6x+16x^{2}=13x^{2}$$

Subtract 13x^{2} from both sides.

$$-6x+16x^{2}-13x^{2}=0$$

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

$$-6x+3x^{2}=0$$

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.

$$3x^{2}-6x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -6 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}}}{2\times 3}$$

Take the square root of \left(-6\right)^{2}.

$$x=\frac{-\left(-6\right)±6}{2\times 3}$$

The opposite of -6 is 6.

$$x=\frac{6±6}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{6±6}{6}$$

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

$$x=\frac{12}{6}$$

Divide 12 by 6.

$$x=2$$

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

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

Divide 0 by 6.

$$x=0$$

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(1+4x\right)^{2}.

$$1+8x+16x^{2}=\left(1+x\right)\left(13x+1\right)$$

Use the distributive property to multiply 1+x by 13x+1 and combine like terms.

$$1+8x+16x^{2}=14x+1+13x^{2}$$

Subtract 14x from both sides.

$$1+8x+16x^{2}-14x=1+13x^{2}$$

Combine 8x and -14x to get -6x.

$$1-6x+16x^{2}=1+13x^{2}$$

Subtract 13x^{2} from both sides.

$$1-6x+16x^{2}-13x^{2}=1$$

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

$$1-6x+3x^{2}=1$$

Subtract 1 from both sides.

$$-6x+3x^{2}=1-1$$

Subtract 1 from 1 to get 0.

$$-6x+3x^{2}=0$$

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.

$$3x^{2}-6x=0$$

Divide both sides by 3.

$$\frac{3x^{2}-6x}{3}=\frac{0}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\left(-\frac{6}{3}\right)x=\frac{0}{3}$$

Divide -6 by 3.

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

Divide 0 by 3.

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

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.

$$x=2 x=0$$