Solve 5x^2-9x^2-32x-12=0

$$5 x ^ { 2 } - 9 x ^ { 2 } - 32 x - 12 = 0$$

Solve for x (complex solution)

$$x=\sqrt{13}-4\approx -0.394448725$$

$$x=-\left(\sqrt{13}+4\right)\approx -7.605551275$$

Steps Using the Quadratic Formula

Combine 5x^{2} and -9x^{2} to get -4x^{2}.

$$-4x^{2}-32x-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, -32 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\left(-4\right)\left(-12\right)}}{2\left(-4\right)}$$

Square -32.

$$x=\frac{-\left(-32\right)±\sqrt{1024-4\left(-4\right)\left(-12\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-\left(-32\right)±\sqrt{1024+16\left(-12\right)}}{2\left(-4\right)}$$

Multiply 16 times -12.

$$x=\frac{-\left(-32\right)±\sqrt{1024-192}}{2\left(-4\right)}$$

Add 1024 to -192.

$$x=\frac{-\left(-32\right)±\sqrt{832}}{2\left(-4\right)}$$

Take the square root of 832.

$$x=\frac{-\left(-32\right)±8\sqrt{13}}{2\left(-4\right)}$$

The opposite of -32 is 32.

$$x=\frac{32±8\sqrt{13}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{32±8\sqrt{13}}{-8}$$

Now solve the equation x=\frac{32±8\sqrt{13}}{-8} when ± is plus. Add 32 to 8\sqrt{13}.

$$x=\frac{8\sqrt{13}+32}{-8}$$

Divide 32+8\sqrt{13} by -8.

$$x=-\left(\sqrt{13}+4\right)$$

Now solve the equation x=\frac{32±8\sqrt{13}}{-8} when ± is minus. Subtract 8\sqrt{13} from 32.

$$x=\frac{32-8\sqrt{13}}{-8}$$

Divide 32-8\sqrt{13} by -8.

$$x=\sqrt{13}-4$$

The equation is now solved.

$$x=-\left(\sqrt{13}+4\right) x=\sqrt{13}-4$$

Steps for Completing the Square

Combine 5x^{2} and -9x^{2} to get -4x^{2}.

$$-4x^{2}-32x-12=0$$

Add 12 to both sides. Anything plus zero gives itself.

$$-4x^{2}-32x=12$$

Divide both sides by -4.

$$\frac{-4x^{2}-32x}{-4}=\frac{12}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\left(-\frac{32}{-4}\right)x=\frac{12}{-4}$$

Divide -32 by -4.

$$x^{2}+8x=\frac{12}{-4}$$

Divide 12 by -4.

$$x^{2}+8x=-3$$

Divide 8, the coefficient of the x term, by 2 to get 4. Then add the square of 4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+8x+4^{2}=-3+4^{2}$$

Square 4.

$$x^{2}+8x+16=-3+16$$

Add -3 to 16.

$$x^{2}+8x+16=13$$

Factor x^{2}+8x+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+4\right)^{2}=13$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{13}$$

Simplify.

$$x+4=\sqrt{13} x+4=-\sqrt{13}$$

Subtract 4 from both sides of the equation.

$$x=\sqrt{13}-4 x=-\sqrt{13}-4$$

Solve for x

$$x=\sqrt{13}-4\approx -0.394448725$$

$$x=-\sqrt{13}-4\approx -7.605551275$$

Steps Using the Quadratic Formula

Combine 5x^{2} and -9x^{2} to get -4x^{2}.

$$-4x^{2}-32x-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, -32 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\left(-4\right)\left(-12\right)}}{2\left(-4\right)}$$

Square -32.

$$x=\frac{-\left(-32\right)±\sqrt{1024-4\left(-4\right)\left(-12\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-\left(-32\right)±\sqrt{1024+16\left(-12\right)}}{2\left(-4\right)}$$

Multiply 16 times -12.

$$x=\frac{-\left(-32\right)±\sqrt{1024-192}}{2\left(-4\right)}$$

Add 1024 to -192.

$$x=\frac{-\left(-32\right)±\sqrt{832}}{2\left(-4\right)}$$

Take the square root of 832.

$$x=\frac{-\left(-32\right)±8\sqrt{13}}{2\left(-4\right)}$$

The opposite of -32 is 32.

$$x=\frac{32±8\sqrt{13}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{32±8\sqrt{13}}{-8}$$

Now solve the equation x=\frac{32±8\sqrt{13}}{-8} when ± is plus. Add 32 to 8\sqrt{13}.

$$x=\frac{8\sqrt{13}+32}{-8}$$

Divide 32+8\sqrt{13} by -8.

$$x=-\left(\sqrt{13}+4\right)$$

Now solve the equation x=\frac{32±8\sqrt{13}}{-8} when ± is minus. Subtract 8\sqrt{13} from 32.

$$x=\frac{32-8\sqrt{13}}{-8}$$

Divide 32-8\sqrt{13} by -8.

$$x=\sqrt{13}-4$$

The equation is now solved.

$$x=-\left(\sqrt{13}+4\right) x=\sqrt{13}-4$$

Steps for Completing the Square

Combine 5x^{2} and -9x^{2} to get -4x^{2}.

$$-4x^{2}-32x-12=0$$

Add 12 to both sides. Anything plus zero gives itself.

$$-4x^{2}-32x=12$$

Divide both sides by -4.

$$\frac{-4x^{2}-32x}{-4}=\frac{12}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\left(-\frac{32}{-4}\right)x=\frac{12}{-4}$$

Divide -32 by -4.

$$x^{2}+8x=\frac{12}{-4}$$

Divide 12 by -4.

$$x^{2}+8x=-3$$

Divide 8, the coefficient of the x term, by 2 to get 4. Then add the square of 4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+8x+4^{2}=-3+4^{2}$$

Square 4.

$$x^{2}+8x+16=-3+16$$

Add -3 to 16.

$$x^{2}+8x+16=13$$

Factor x^{2}+8x+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+4\right)^{2}=13$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{13}$$

Simplify.

$$x+4=\sqrt{13} x+4=-\sqrt{13}$$

Subtract 4 from both sides of the equation.

$$x=\sqrt{13}-4 x=-\sqrt{13}-4$$