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

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

Solve for x

$$x=\frac{\sqrt{106}-13}{9}\approx -0.30048554$$

$$x=\frac{-\sqrt{106}-13}{9}\approx -2.588403349$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3x+4\right)^{2}.

$$2\left(9x^{2}+24x+16\right)+4\left(x-3\right)=6$$

Use the distributive property to multiply 2 by 9x^{2}+24x+16.

$$18x^{2}+48x+32+4\left(x-3\right)=6$$

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

$$18x^{2}+48x+32+4x-12=6$$

Combine 48x and 4x to get 52x.

$$18x^{2}+52x+32-12=6$$

Subtract 12 from 32 to get 20.

$$18x^{2}+52x+20=6$$

Subtract 6 from both sides.

$$18x^{2}+52x+20-6=0$$

Subtract 6 from 20 to get 14.

$$18x^{2}+52x+14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 18 for a, 52 for b, and 14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-52±\sqrt{52^{2}-4\times 18\times 14}}{2\times 18}$$

Square 52.

$$x=\frac{-52±\sqrt{2704-4\times 18\times 14}}{2\times 18}$$

Multiply -4 times 18.

$$x=\frac{-52±\sqrt{2704-72\times 14}}{2\times 18}$$

Multiply -72 times 14.

$$x=\frac{-52±\sqrt{2704-1008}}{2\times 18}$$

Add 2704 to -1008.

$$x=\frac{-52±\sqrt{1696}}{2\times 18}$$

Take the square root of 1696.

$$x=\frac{-52±4\sqrt{106}}{2\times 18}$$

Multiply 2 times 18.

$$x=\frac{-52±4\sqrt{106}}{36}$$

Now solve the equation x=\frac{-52±4\sqrt{106}}{36} when ± is plus. Add -52 to 4\sqrt{106}.

$$x=\frac{4\sqrt{106}-52}{36}$$

Divide -52+4\sqrt{106} by 36.

$$x=\frac{\sqrt{106}-13}{9}$$

Now solve the equation x=\frac{-52±4\sqrt{106}}{36} when ± is minus. Subtract 4\sqrt{106} from -52.

$$x=\frac{-4\sqrt{106}-52}{36}$$

Divide -52-4\sqrt{106} by 36.

$$x=\frac{-\sqrt{106}-13}{9}$$

The equation is now solved.

$$x=\frac{\sqrt{106}-13}{9} x=\frac{-\sqrt{106}-13}{9}$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3x+4\right)^{2}.

$$2\left(9x^{2}+24x+16\right)+4\left(x-3\right)=6$$

Use the distributive property to multiply 2 by 9x^{2}+24x+16.

$$18x^{2}+48x+32+4\left(x-3\right)=6$$

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

$$18x^{2}+48x+32+4x-12=6$$

Combine 48x and 4x to get 52x.

$$18x^{2}+52x+32-12=6$$

Subtract 12 from 32 to get 20.

$$18x^{2}+52x+20=6$$

Subtract 20 from both sides.

$$18x^{2}+52x=6-20$$

Subtract 20 from 6 to get -14.

$$18x^{2}+52x=-14$$

Divide both sides by 18.

$$\frac{18x^{2}+52x}{18}=-\frac{14}{18}$$

Dividing by 18 undoes the multiplication by 18.

$$x^{2}+\frac{52}{18}x=-\frac{14}{18}$$

Reduce the fraction \frac{52}{18} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{26}{9}x=-\frac{14}{18}$$

Reduce the fraction \frac{-14}{18} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{26}{9}x=-\frac{7}{9}$$

Divide \frac{26}{9}, the coefficient of the x term, by 2 to get \frac{13}{9}. Then add the square of \frac{13}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{26}{9}x+\left(\frac{13}{9}\right)^{2}=-\frac{7}{9}+\left(\frac{13}{9}\right)^{2}$$

Square \frac{13}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{26}{9}x+\frac{169}{81}=-\frac{7}{9}+\frac{169}{81}$$

Add -\frac{7}{9} to \frac{169}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{26}{9}x+\frac{169}{81}=\frac{106}{81}$$

Factor x^{2}+\frac{26}{9}x+\frac{169}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{13}{9}\right)^{2}=\frac{106}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{13}{9}\right)^{2}}=\sqrt{\frac{106}{81}}$$

Simplify.

$$x+\frac{13}{9}=\frac{\sqrt{106}}{9} x+\frac{13}{9}=-\frac{\sqrt{106}}{9}$$

Subtract \frac{13}{9} from both sides of the equation.

$$x=\frac{\sqrt{106}-13}{9} x=\frac{-\sqrt{106}-13}{9}$$