Solve 0=-6x^2+16x+44

$$0=-6 { x }^{ 2 } +16x+44$$

Solve for x

$$x = \frac{\sqrt{82} + 4}{3} \approx 4.351795046$$

$$x=\frac{4-\sqrt{82}}{3}\approx -1.685128379$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$-6x^{2}+16x+44=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6 for a, 16 for b, and 44 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-6\right)\times 44}}{2\left(-6\right)}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\left(-6\right)\times 44}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{-16±\sqrt{256+24\times 44}}{2\left(-6\right)}$$

Multiply 24 times 44.

$$x=\frac{-16±\sqrt{256+1056}}{2\left(-6\right)}$$

Add 256 to 1056.

$$x=\frac{-16±\sqrt{1312}}{2\left(-6\right)}$$

Take the square root of 1312.

$$x=\frac{-16±4\sqrt{82}}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{-16±4\sqrt{82}}{-12}$$

Now solve the equation x=\frac{-16±4\sqrt{82}}{-12} when ± is plus. Add -16 to 4\sqrt{82}.

$$x=\frac{4\sqrt{82}-16}{-12}$$

Divide -16+4\sqrt{82} by -12.

$$x=\frac{4-\sqrt{82}}{3}$$

Now solve the equation x=\frac{-16±4\sqrt{82}}{-12} when ± is minus. Subtract 4\sqrt{82} from -16.

$$x=\frac{-4\sqrt{82}-16}{-12}$$

Divide -16-4\sqrt{82} by -12.

$$x=\frac{\sqrt{82}+4}{3}$$

The equation is now solved.

$$x=\frac{4-\sqrt{82}}{3} x=\frac{\sqrt{82}+4}{3}$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$-6x^{2}+16x+44=0$$

Subtract 44 from both sides. Anything subtracted from zero gives its negation.

$$-6x^{2}+16x=-44$$

Divide both sides by -6.

$$\frac{-6x^{2}+16x}{-6}=-\frac{44}{-6}$$

Dividing by -6 undoes the multiplication by -6.

$$x^{2}+\frac{16}{-6}x=-\frac{44}{-6}$$

Reduce the fraction \frac{16}{-6} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{8}{3}x=-\frac{44}{-6}$$

Reduce the fraction \frac{-44}{-6} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{8}{3}x=\frac{22}{3}$$

Divide -\frac{8}{3}, the coefficient of the x term, by 2 to get -\frac{4}{3}. Then add the square of -\frac{4}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{8}{3}x+\left(-\frac{4}{3}\right)^{2}=\frac{22}{3}+\left(-\frac{4}{3}\right)^{2}$$

Square -\frac{4}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{8}{3}x+\frac{16}{9}=\frac{22}{3}+\frac{16}{9}$$

Add \frac{22}{3} to \frac{16}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{8}{3}x+\frac{16}{9}=\frac{82}{9}$$

Factor x^{2}-\frac{8}{3}x+\frac{16}{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}.

$$\left(x-\frac{4}{3}\right)^{2}=\frac{82}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{4}{3}\right)^{2}}=\sqrt{\frac{82}{9}}$$

Simplify.

$$x-\frac{4}{3}=\frac{\sqrt{82}}{3} x-\frac{4}{3}=-\frac{\sqrt{82}}{3}$$

Add \frac{4}{3} to both sides of the equation.

$$x=\frac{\sqrt{82}+4}{3} x=\frac{4-\sqrt{82}}{3}$$