Solve 2x^2+6x^2+7x+60=0

$$2 { x }^{ 2 } +6 { x }^{ 2 } +7x+60 = 0$$

Solve for x (complex solution)

$$x=\frac{-7+\sqrt{1871}i}{16}\approx -0.4375+2.703441094i$$

$$x=\frac{-\sqrt{1871}i-7}{16}\approx -0.4375-2.703441094i$$

Steps Using the Quadratic Formula

Combine 2x^{2} and 6x^{2} to get 8x^{2}.

$$8x^{2}+7x+60=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, 7 for b, and 60 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\times 8\times 60}}{2\times 8}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\times 8\times 60}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-7±\sqrt{49-32\times 60}}{2\times 8}$$

Multiply -32 times 60.

$$x=\frac{-7±\sqrt{49-1920}}{2\times 8}$$

Add 49 to -1920.

$$x=\frac{-7±\sqrt{-1871}}{2\times 8}$$

Take the square root of -1871.

$$x=\frac{-7±\sqrt{1871}i}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{-7±\sqrt{1871}i}{16}$$

Now solve the equation x=\frac{-7±\sqrt{1871}i}{16} when ± is plus. Add -7 to i\sqrt{1871}.

$$x=\frac{-7+\sqrt{1871}i}{16}$$

Now solve the equation x=\frac{-7±\sqrt{1871}i}{16} when ± is minus. Subtract i\sqrt{1871} from -7.

$$x=\frac{-\sqrt{1871}i-7}{16}$$

The equation is now solved.

$$x=\frac{-7+\sqrt{1871}i}{16} x=\frac{-\sqrt{1871}i-7}{16}$$

Steps for Completing the Square

Combine 2x^{2} and 6x^{2} to get 8x^{2}.

$$8x^{2}+7x+60=0$$

Subtract 60 from both sides. Anything subtracted from zero gives its negation.

$$8x^{2}+7x=-60$$

Divide both sides by 8.

$$\frac{8x^{2}+7x}{8}=-\frac{60}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$x^{2}+\frac{7}{8}x=-\frac{60}{8}$$

Reduce the fraction \frac{-60}{8} to lowest terms by extracting and canceling out 4.

$$x^{2}+\frac{7}{8}x=-\frac{15}{2}$$

Divide \frac{7}{8}, the coefficient of the x term, by 2 to get \frac{7}{16}. Then add the square of \frac{7}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{7}{8}x+\left(\frac{7}{16}\right)^{2}=-\frac{15}{2}+\left(\frac{7}{16}\right)^{2}$$

Square \frac{7}{16} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{7}{8}x+\frac{49}{256}=-\frac{15}{2}+\frac{49}{256}$$

Add -\frac{15}{2} to \frac{49}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{7}{8}x+\frac{49}{256}=-\frac{1871}{256}$$

Factor x^{2}+\frac{7}{8}x+\frac{49}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{7}{16}\right)^{2}=-\frac{1871}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{16}\right)^{2}}=\sqrt{-\frac{1871}{256}}$$

Simplify.

$$x+\frac{7}{16}=\frac{\sqrt{1871}i}{16} x+\frac{7}{16}=-\frac{\sqrt{1871}i}{16}$$

Subtract \frac{7}{16} from both sides of the equation.

$$x=\frac{-7+\sqrt{1871}i}{16} x=\frac{-\sqrt{1871}i-7}{16}$$