Solve 4(2x+1)^2/3=36

$$4 ( 2 x + 1 ) ^ { 2 } / 3 = 36$$

Solve for x

$$x = \frac{3 \sqrt{3} - 1}{2} \approx 2.098076211$$

$$x=\frac{-3\sqrt{3}-1}{2}\approx -3.098076211$$

Steps Using the Quadratic Formula

Multiply both sides by 3.

$$4\left(2x+1\right)^{2}=36\times 3$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+1\right)^{2}.

$$4\left(4x^{2}+4x+1\right)=36\times 3$$

Use the distributive property to multiply 4 by 4x^{2}+4x+1.

$$16x^{2}+16x+4=36\times 3$$

Multiply 36 and 3 to get 108.

$$16x^{2}+16x+4=108$$

Subtract 108 from both sides.

$$16x^{2}+16x+4-108=0$$

Subtract 108 from 4 to get -104.

$$16x^{2}+16x-104=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, 16 for b, and -104 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\times 16\left(-104\right)}}{2\times 16}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\times 16\left(-104\right)}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-16±\sqrt{256-64\left(-104\right)}}{2\times 16}$$

Multiply -64 times -104.

$$x=\frac{-16±\sqrt{256+6656}}{2\times 16}$$

Add 256 to 6656.

$$x=\frac{-16±\sqrt{6912}}{2\times 16}$$

Take the square root of 6912.

$$x=\frac{-16±48\sqrt{3}}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{-16±48\sqrt{3}}{32}$$

Now solve the equation x=\frac{-16±48\sqrt{3}}{32} when ± is plus. Add -16 to 48\sqrt{3}.

$$x=\frac{48\sqrt{3}-16}{32}$$

Divide 48\sqrt{3}-16 by 32.

$$x=\frac{3\sqrt{3}-1}{2}$$

Now solve the equation x=\frac{-16±48\sqrt{3}}{32} when ± is minus. Subtract 48\sqrt{3} from -16.

$$x=\frac{-48\sqrt{3}-16}{32}$$

Divide -16-48\sqrt{3} by 32.

$$x=\frac{-3\sqrt{3}-1}{2}$$

The equation is now solved.

$$x=\frac{3\sqrt{3}-1}{2} x=\frac{-3\sqrt{3}-1}{2}$$

Steps for Completing the Square

Multiply both sides by 3.

$$4\left(2x+1\right)^{2}=36\times 3$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+1\right)^{2}.

$$4\left(4x^{2}+4x+1\right)=36\times 3$$

Use the distributive property to multiply 4 by 4x^{2}+4x+1.

$$16x^{2}+16x+4=36\times 3$$

Multiply 36 and 3 to get 108.

$$16x^{2}+16x+4=108$$

Subtract 4 from both sides.

$$16x^{2}+16x=108-4$$

Subtract 4 from 108 to get 104.

$$16x^{2}+16x=104$$

Divide both sides by 16.

$$\frac{16x^{2}+16x}{16}=\frac{104}{16}$$

Dividing by 16 undoes the multiplication by 16.

$$x^{2}+\frac{16}{16}x=\frac{104}{16}$$

Divide 16 by 16.

$$x^{2}+x=\frac{104}{16}$$

Reduce the fraction \frac{104}{16} to lowest terms by extracting and canceling out 8.

$$x^{2}+x=\frac{13}{2}$$

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

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

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+x+\frac{1}{4}=\frac{13}{2}+\frac{1}{4}$$

Add \frac{13}{2} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+x+\frac{1}{4}=\frac{27}{4}$$

Factor x^{2}+x+\frac{1}{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}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{27}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{27}{4}}$$

Simplify.

$$x+\frac{1}{2}=\frac{3\sqrt{3}}{2} x+\frac{1}{2}=-\frac{3\sqrt{3}}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=\frac{3\sqrt{3}-1}{2} x=\frac{-3\sqrt{3}-1}{2}$$