Solve 15x^2+28x+12=9x^2+4+1

$$15 x ^ { 2 } + 28 x + 12 = 9 x ^ { 2 } + 4 + 1$$

Solve for x

$$x=\frac{\sqrt{154}}{6}-\frac{7}{3}\approx -0.265054392$$

$$x=-\frac{\sqrt{154}}{6}-\frac{7}{3}\approx -4.401612274$$

Steps Using the Quadratic Formula

Add 4 and 1 to get 5.

$$15x^{2}+28x+12=9x^{2}+5$$

Subtract 9x^{2} from both sides.

$$15x^{2}+28x+12-9x^{2}=5$$

Combine 15x^{2} and -9x^{2} to get 6x^{2}.

$$6x^{2}+28x+12=5$$

Subtract 5 from both sides.

$$6x^{2}+28x+12-5=0$$

Subtract 5 from 12 to get 7.

$$6x^{2}+28x+7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 28 for b, and 7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-28±\sqrt{28^{2}-4\times 6\times 7}}{2\times 6}$$

Square 28.

$$x=\frac{-28±\sqrt{784-4\times 6\times 7}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-28±\sqrt{784-24\times 7}}{2\times 6}$$

Multiply -24 times 7.

$$x=\frac{-28±\sqrt{784-168}}{2\times 6}$$

Add 784 to -168.

$$x=\frac{-28±\sqrt{616}}{2\times 6}$$

Take the square root of 616.

$$x=\frac{-28±2\sqrt{154}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-28±2\sqrt{154}}{12}$$

Now solve the equation x=\frac{-28±2\sqrt{154}}{12} when ± is plus. Add -28 to 2\sqrt{154}.

$$x=\frac{2\sqrt{154}-28}{12}$$

Divide -28+2\sqrt{154} by 12.

$$x=\frac{\sqrt{154}}{6}-\frac{7}{3}$$

Now solve the equation x=\frac{-28±2\sqrt{154}}{12} when ± is minus. Subtract 2\sqrt{154} from -28.

$$x=\frac{-2\sqrt{154}-28}{12}$$

Divide -28-2\sqrt{154} by 12.

$$x=-\frac{\sqrt{154}}{6}-\frac{7}{3}$$

The equation is now solved.

$$x=\frac{\sqrt{154}}{6}-\frac{7}{3} x=-\frac{\sqrt{154}}{6}-\frac{7}{3}$$

Steps for Completing the Square

Add 4 and 1 to get 5.

$$15x^{2}+28x+12=9x^{2}+5$$

Subtract 9x^{2} from both sides.

$$15x^{2}+28x+12-9x^{2}=5$$

Combine 15x^{2} and -9x^{2} to get 6x^{2}.

$$6x^{2}+28x+12=5$$

Subtract 12 from both sides.

$$6x^{2}+28x=5-12$$

Subtract 12 from 5 to get -7.

$$6x^{2}+28x=-7$$

Divide both sides by 6.

$$\frac{6x^{2}+28x}{6}=-\frac{7}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\frac{28}{6}x=-\frac{7}{6}$$

Reduce the fraction \frac{28}{6} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{14}{3}x=-\frac{7}{6}$$

Divide \frac{14}{3}, the coefficient of the x term, by 2 to get \frac{7}{3}. Then add the square of \frac{7}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{14}{3}x+\left(\frac{7}{3}\right)^{2}=-\frac{7}{6}+\left(\frac{7}{3}\right)^{2}$$

Square \frac{7}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{14}{3}x+\frac{49}{9}=-\frac{7}{6}+\frac{49}{9}$$

Add -\frac{7}{6} to \frac{49}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{14}{3}x+\frac{49}{9}=\frac{77}{18}$$

Factor x^{2}+\frac{14}{3}x+\frac{49}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{7}{3}\right)^{2}=\frac{77}{18}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{3}\right)^{2}}=\sqrt{\frac{77}{18}}$$

Simplify.

$$x+\frac{7}{3}=\frac{\sqrt{154}}{6} x+\frac{7}{3}=-\frac{\sqrt{154}}{6}$$

Subtract \frac{7}{3} from both sides of the equation.

$$x=\frac{\sqrt{154}}{6}-\frac{7}{3} x=-\frac{\sqrt{154}}{6}-\frac{7}{3}$$