Solve 50=-16x^2+216x-672

$$50 = - 16 x ^ { 2 } + 216 x - 672$$

Solve for x

$$x = \frac{\sqrt{7} + 27}{4} \approx 7.411437828$$

$$x = \frac{27 - \sqrt{7}}{4} \approx 6.088562172$$

Steps Using the Quadratic Formula

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

$$-16x^{2}+216x-672=50$$

Subtract 50 from both sides.

$$-16x^{2}+216x-672-50=0$$

Subtract 50 from -672 to get -722.

$$-16x^{2}+216x-722=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 216 for b, and -722 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-216±\sqrt{216^{2}-4\left(-16\right)\left(-722\right)}}{2\left(-16\right)}$$

Square 216.

$$x=\frac{-216±\sqrt{46656-4\left(-16\right)\left(-722\right)}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$x=\frac{-216±\sqrt{46656+64\left(-722\right)}}{2\left(-16\right)}$$

Multiply 64 times -722.

$$x=\frac{-216±\sqrt{46656-46208}}{2\left(-16\right)}$$

Add 46656 to -46208.

$$x=\frac{-216±\sqrt{448}}{2\left(-16\right)}$$

Take the square root of 448.

$$x=\frac{-216±8\sqrt{7}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$x=\frac{-216±8\sqrt{7}}{-32}$$

Now solve the equation x=\frac{-216±8\sqrt{7}}{-32} when ± is plus. Add -216 to 8\sqrt{7}.

$$x=\frac{8\sqrt{7}-216}{-32}$$

Divide -216+8\sqrt{7} by -32.

$$x=\frac{27-\sqrt{7}}{4}$$

Now solve the equation x=\frac{-216±8\sqrt{7}}{-32} when ± is minus. Subtract 8\sqrt{7} from -216.

$$x=\frac{-8\sqrt{7}-216}{-32}$$

Divide -216-8\sqrt{7} by -32.

$$x=\frac{\sqrt{7}+27}{4}$$

The equation is now solved.

$$x=\frac{27-\sqrt{7}}{4} x=\frac{\sqrt{7}+27}{4}$$

Steps for Completing the Square

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

$$-16x^{2}+216x-672=50$$

Add 672 to both sides.

$$-16x^{2}+216x=50+672$$

Add 50 and 672 to get 722.

$$-16x^{2}+216x=722$$

Divide both sides by -16.

$$\frac{-16x^{2}+216x}{-16}=\frac{722}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$x^{2}+\frac{216}{-16}x=\frac{722}{-16}$$

Reduce the fraction \frac{216}{-16} to lowest terms by extracting and canceling out 8.

$$x^{2}-\frac{27}{2}x=\frac{722}{-16}$$

Reduce the fraction \frac{722}{-16} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{27}{2}x=-\frac{361}{8}$$

Divide -\frac{27}{2}, the coefficient of the x term, by 2 to get -\frac{27}{4}. Then add the square of -\frac{27}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{27}{2}x+\left(-\frac{27}{4}\right)^{2}=-\frac{361}{8}+\left(-\frac{27}{4}\right)^{2}$$

Square -\frac{27}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{27}{2}x+\frac{729}{16}=-\frac{361}{8}+\frac{729}{16}$$

Add -\frac{361}{8} to \frac{729}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{27}{2}x+\frac{729}{16}=\frac{7}{16}$$

Factor x^{2}-\frac{27}{2}x+\frac{729}{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}.

$$\left(x-\frac{27}{4}\right)^{2}=\frac{7}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{27}{4}\right)^{2}}=\sqrt{\frac{7}{16}}$$

Simplify.

$$x-\frac{27}{4}=\frac{\sqrt{7}}{4} x-\frac{27}{4}=-\frac{\sqrt{7}}{4}$$

Add \frac{27}{4} to both sides of the equation.

$$x=\frac{\sqrt{7}+27}{4} x=\frac{27-\sqrt{7}}{4}$$