Solve 12x^2+3xleft(1+xright)+9=90

$$12 { x }^{ 2 } +3x \left( 1+x \right) +9 = 90$$

Solve for x

$$x = \frac{\sqrt{541} - 1}{10} \approx 2.22594067$$

$$x=\frac{-\sqrt{541}-1}{10}\approx -2.42594067$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3x by 1+x.

$$12x^{2}+3x+3x^{2}+9=90$$

Combine 12x^{2} and 3x^{2} to get 15x^{2}.

$$15x^{2}+3x+9=90$$

Subtract 90 from both sides.

$$15x^{2}+3x+9-90=0$$

Subtract 90 from 9 to get -81.

$$15x^{2}+3x-81=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 15 for a, 3 for b, and -81 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\times 15\left(-81\right)}}{2\times 15}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\times 15\left(-81\right)}}{2\times 15}$$

Multiply -4 times 15.

$$x=\frac{-3±\sqrt{9-60\left(-81\right)}}{2\times 15}$$

Multiply -60 times -81.

$$x=\frac{-3±\sqrt{9+4860}}{2\times 15}$$

Add 9 to 4860.

$$x=\frac{-3±\sqrt{4869}}{2\times 15}$$

Take the square root of 4869.

$$x=\frac{-3±3\sqrt{541}}{2\times 15}$$

Multiply 2 times 15.

$$x=\frac{-3±3\sqrt{541}}{30}$$

Now solve the equation x=\frac{-3±3\sqrt{541}}{30} when ± is plus. Add -3 to 3\sqrt{541}.

$$x=\frac{3\sqrt{541}-3}{30}$$

Divide -3+3\sqrt{541} by 30.

$$x=\frac{\sqrt{541}-1}{10}$$

Now solve the equation x=\frac{-3±3\sqrt{541}}{30} when ± is minus. Subtract 3\sqrt{541} from -3.

$$x=\frac{-3\sqrt{541}-3}{30}$$

Divide -3-3\sqrt{541} by 30.

$$x=\frac{-\sqrt{541}-1}{10}$$

The equation is now solved.

$$x=\frac{\sqrt{541}-1}{10} x=\frac{-\sqrt{541}-1}{10}$$

Steps for Completing the Square

Use the distributive property to multiply 3x by 1+x.

$$12x^{2}+3x+3x^{2}+9=90$$

Combine 12x^{2} and 3x^{2} to get 15x^{2}.

$$15x^{2}+3x+9=90$$

Subtract 9 from both sides.

$$15x^{2}+3x=90-9$$

Subtract 9 from 90 to get 81.

$$15x^{2}+3x=81$$

Divide both sides by 15.

$$\frac{15x^{2}+3x}{15}=\frac{81}{15}$$

Dividing by 15 undoes the multiplication by 15.

$$x^{2}+\frac{3}{15}x=\frac{81}{15}$$

Reduce the fraction \frac{3}{15} to lowest terms by extracting and canceling out 3.

$$x^{2}+\frac{1}{5}x=\frac{81}{15}$$

Reduce the fraction \frac{81}{15} to lowest terms by extracting and canceling out 3.

$$x^{2}+\frac{1}{5}x=\frac{27}{5}$$

Divide \frac{1}{5}, the coefficient of the x term, by 2 to get \frac{1}{10}. Then add the square of \frac{1}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{5}x+\left(\frac{1}{10}\right)^{2}=\frac{27}{5}+\left(\frac{1}{10}\right)^{2}$$

Square \frac{1}{10} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{5}x+\frac{1}{100}=\frac{27}{5}+\frac{1}{100}$$

Add \frac{27}{5} to \frac{1}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{5}x+\frac{1}{100}=\frac{541}{100}$$

Factor x^{2}+\frac{1}{5}x+\frac{1}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{10}\right)^{2}=\frac{541}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{10}\right)^{2}}=\sqrt{\frac{541}{100}}$$

Simplify.

$$x+\frac{1}{10}=\frac{\sqrt{541}}{10} x+\frac{1}{10}=-\frac{\sqrt{541}}{10}$$

Subtract \frac{1}{10} from both sides of the equation.

$$x=\frac{\sqrt{541}-1}{10} x=\frac{-\sqrt{541}-1}{10}$$