Solve (4)4(2x+3)^2=(-3)^2

$$( 4 ) 4 ( 2 x + 3 ) ^ { 2 } = ( - 3 ) ^ { 2 }$$

Solve for x

$$x = -\frac{9}{8} = -1\frac{1}{8} = -1.125$$

$$x = -\frac{15}{8} = -1\frac{7}{8} = -1.875$$

Steps Using the Quadratic Formula

Multiply 4 and 4 to get 16.

$$16\left(2x+3\right)^{2}=\left(-3\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+3\right)^{2}.

$$16\left(4x^{2}+12x+9\right)=\left(-3\right)^{2}$$

Use the distributive property to multiply 16 by 4x^{2}+12x+9.

$$64x^{2}+192x+144=\left(-3\right)^{2}$$

Calculate -3 to the power of 2 and get 9.

$$64x^{2}+192x+144=9$$

Subtract 9 from both sides.

$$64x^{2}+192x+144-9=0$$

Subtract 9 from 144 to get 135.

$$64x^{2}+192x+135=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 64 for a, 192 for b, and 135 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-192±\sqrt{192^{2}-4\times 64\times 135}}{2\times 64}$$

Square 192.

$$x=\frac{-192±\sqrt{36864-4\times 64\times 135}}{2\times 64}$$

Multiply -4 times 64.

$$x=\frac{-192±\sqrt{36864-256\times 135}}{2\times 64}$$

Multiply -256 times 135.

$$x=\frac{-192±\sqrt{36864-34560}}{2\times 64}$$

Add 36864 to -34560.

$$x=\frac{-192±\sqrt{2304}}{2\times 64}$$

Take the square root of 2304.

$$x=\frac{-192±48}{2\times 64}$$

Multiply 2 times 64.

$$x=\frac{-192±48}{128}$$

Now solve the equation x=\frac{-192±48}{128} when ± is plus. Add -192 to 48.

$$x=-\frac{144}{128}$$

Reduce the fraction \frac{-144}{128} to lowest terms by extracting and canceling out 16.

$$x=-\frac{9}{8}$$

Now solve the equation x=\frac{-192±48}{128} when ± is minus. Subtract 48 from -192.

$$x=-\frac{240}{128}$$

Reduce the fraction \frac{-240}{128} to lowest terms by extracting and canceling out 16.

$$x=-\frac{15}{8}$$

The equation is now solved.

$$x=-\frac{9}{8} x=-\frac{15}{8}$$

Steps for Completing the Square

Multiply 4 and 4 to get 16.

$$16\left(2x+3\right)^{2}=\left(-3\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+3\right)^{2}.

$$16\left(4x^{2}+12x+9\right)=\left(-3\right)^{2}$$

Use the distributive property to multiply 16 by 4x^{2}+12x+9.

$$64x^{2}+192x+144=\left(-3\right)^{2}$$

Calculate -3 to the power of 2 and get 9.

$$64x^{2}+192x+144=9$$

Subtract 144 from both sides.

$$64x^{2}+192x=9-144$$

Subtract 144 from 9 to get -135.

$$64x^{2}+192x=-135$$

Divide both sides by 64.

$$\frac{64x^{2}+192x}{64}=-\frac{135}{64}$$

Dividing by 64 undoes the multiplication by 64.

$$x^{2}+\frac{192}{64}x=-\frac{135}{64}$$

Divide 192 by 64.

$$x^{2}+3x=-\frac{135}{64}$$

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.

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=-\frac{135}{64}+\left(\frac{3}{2}\right)^{2}$$

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

$$x^{2}+3x+\frac{9}{4}=-\frac{135}{64}+\frac{9}{4}$$

Add -\frac{135}{64} to \frac{9}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+3x+\frac{9}{4}=\frac{9}{64}$$

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}.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{9}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{9}{64}}$$

Simplify.

$$x+\frac{3}{2}=\frac{3}{8} x+\frac{3}{2}=-\frac{3}{8}$$

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

$$x=-\frac{9}{8} x=-\frac{15}{8}$$