Solve 3x^2-3x(4x+6)+(4x+6)^2=12

$$3 x ^ { 2 } - 3 x ( 4 x + 6 ) + ( 4 x + 6 ) ^ { 2 } = 12$$

Solve for x

$$x=\frac{\sqrt{57}-15}{7}\approx -1.064309366$$

$$x=\frac{-\sqrt{57}-15}{7}\approx -3.221404919$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4x+6\right)^{2}.

$$3x^{2}-3x\left(4x+6\right)+16x^{2}+48x+36=12$$

Subtract 12 from both sides.

$$3x^{2}-3x\left(4x+6\right)+16x^{2}+48x+36-12=0$$

Subtract 12 from 36 to get 24.

$$3x^{2}-3x\left(4x+6\right)+16x^{2}+48x+24=0$$

Use the distributive property to multiply -3x by 4x+6.

$$3x^{2}-12x^{2}-18x+16x^{2}+48x+24=0$$

Combine 3x^{2} and -12x^{2} to get -9x^{2}.

$$-9x^{2}-18x+16x^{2}+48x+24=0$$

Combine -9x^{2} and 16x^{2} to get 7x^{2}.

$$7x^{2}-18x+48x+24=0$$

Combine -18x and 48x to get 30x.

$$7x^{2}+30x+24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, 30 for b, and 24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-30±\sqrt{30^{2}-4\times 7\times 24}}{2\times 7}$$

Square 30.

$$x=\frac{-30±\sqrt{900-4\times 7\times 24}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-30±\sqrt{900-28\times 24}}{2\times 7}$$

Multiply -28 times 24.

$$x=\frac{-30±\sqrt{900-672}}{2\times 7}$$

Add 900 to -672.

$$x=\frac{-30±\sqrt{228}}{2\times 7}$$

Take the square root of 228.

$$x=\frac{-30±2\sqrt{57}}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{-30±2\sqrt{57}}{14}$$

Now solve the equation x=\frac{-30±2\sqrt{57}}{14} when ± is plus. Add -30 to 2\sqrt{57}.

$$x=\frac{2\sqrt{57}-30}{14}$$

Divide -30+2\sqrt{57} by 14.

$$x=\frac{\sqrt{57}-15}{7}$$

Now solve the equation x=\frac{-30±2\sqrt{57}}{14} when ± is minus. Subtract 2\sqrt{57} from -30.

$$x=\frac{-2\sqrt{57}-30}{14}$$

Divide -30-2\sqrt{57} by 14.

$$x=\frac{-\sqrt{57}-15}{7}$$

The equation is now solved.

$$x=\frac{\sqrt{57}-15}{7} x=\frac{-\sqrt{57}-15}{7}$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4x+6\right)^{2}.

$$3x^{2}-3x\left(4x+6\right)+16x^{2}+48x+36=12$$

Subtract 36 from both sides.

$$3x^{2}-3x\left(4x+6\right)+16x^{2}+48x=12-36$$

Subtract 36 from 12 to get -24.

$$3x^{2}-3x\left(4x+6\right)+16x^{2}+48x=-24$$

Use the distributive property to multiply -3x by 4x+6.

$$3x^{2}-12x^{2}-18x+16x^{2}+48x=-24$$

Combine 3x^{2} and -12x^{2} to get -9x^{2}.

$$-9x^{2}-18x+16x^{2}+48x=-24$$

Combine -9x^{2} and 16x^{2} to get 7x^{2}.

$$7x^{2}-18x+48x=-24$$

Combine -18x and 48x to get 30x.

$$7x^{2}+30x=-24$$

Divide both sides by 7.

$$\frac{7x^{2}+30x}{7}=-\frac{24}{7}$$

Dividing by 7 undoes the multiplication by 7.

$$x^{2}+\frac{30}{7}x=-\frac{24}{7}$$

Divide \frac{30}{7}, the coefficient of the x term, by 2 to get \frac{15}{7}. Then add the square of \frac{15}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{30}{7}x+\left(\frac{15}{7}\right)^{2}=-\frac{24}{7}+\left(\frac{15}{7}\right)^{2}$$

Square \frac{15}{7} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{30}{7}x+\frac{225}{49}=-\frac{24}{7}+\frac{225}{49}$$

Add -\frac{24}{7} to \frac{225}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{30}{7}x+\frac{225}{49}=\frac{57}{49}$$

Factor x^{2}+\frac{30}{7}x+\frac{225}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{15}{7}\right)^{2}=\frac{57}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{15}{7}\right)^{2}}=\sqrt{\frac{57}{49}}$$

Simplify.

$$x+\frac{15}{7}=\frac{\sqrt{57}}{7} x+\frac{15}{7}=-\frac{\sqrt{57}}{7}$$

Subtract \frac{15}{7} from both sides of the equation.

$$x=\frac{\sqrt{57}-15}{7} x=\frac{-\sqrt{57}-15}{7}$$