Solve (2+2x)(4+2x)=24

$$( 2 + 2 x ) ( 4 + 2 x ) = 24$$

Solve for x

$$x=-4$$

$$x=1$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2+2x by 4+2x and combine like terms.

$$8+12x+4x^{2}=24$$

Subtract 24 from both sides.

$$8+12x+4x^{2}-24=0$$

Subtract 24 from 8 to get -16.

$$-16+12x+4x^{2}=0$$

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.

$$4x^{2}+12x-16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 12 for b, and -16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\times 4\left(-16\right)}}{2\times 4}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\times 4\left(-16\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-12±\sqrt{144-16\left(-16\right)}}{2\times 4}$$

Multiply -16 times -16.

$$x=\frac{-12±\sqrt{144+256}}{2\times 4}$$

Add 144 to 256.

$$x=\frac{-12±\sqrt{400}}{2\times 4}$$

Take the square root of 400.

$$x=\frac{-12±20}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-12±20}{8}$$

Now solve the equation x=\frac{-12±20}{8} when ± is plus. Add -12 to 20.

$$x=\frac{8}{8}$$

Divide 8 by 8.

$$x=1$$

Now solve the equation x=\frac{-12±20}{8} when ± is minus. Subtract 20 from -12.

$$x=-\frac{32}{8}$$

Divide -32 by 8.

$$x=-4$$

The equation is now solved.

$$x=1 x=-4$$

Steps for Completing the Square

Use the distributive property to multiply 2+2x by 4+2x and combine like terms.

$$8+12x+4x^{2}=24$$

Subtract 8 from both sides.

$$12x+4x^{2}=24-8$$

Subtract 8 from 24 to get 16.

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

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.

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

Divide both sides by 4.

$$\frac{4x^{2}+12x}{4}=\frac{16}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{12}{4}x=\frac{16}{4}$$

Divide 12 by 4.

$$x^{2}+3x=\frac{16}{4}$$

Divide 16 by 4.

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

Divide 3, the coefficient of the x term, by 2 to get \frac{3}{2}. Then add the square of \frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=4+\left(\frac{3}{2}\right)^{2}$$

Square \frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+3x+\frac{9}{4}=4+\frac{9}{4}$$

Add 4 to \frac{9}{4}.

$$x^{2}+3x+\frac{9}{4}=\frac{25}{4}$$

Factor x^{2}+3x+\frac{9}{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}.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Simplify.

$$x+\frac{3}{2}=\frac{5}{2} x+\frac{3}{2}=-\frac{5}{2}$$

Subtract \frac{3}{2} from both sides of the equation.

$$x=1 x=-4$$