Solve 0=-16x^2+17x+7

$$0=-16 { x }^{ 2 } +17x+7$$

Solve for x

$$x = \frac{\sqrt{737} + 17}{32} \approx 1.379616998$$

$$x=\frac{17-\sqrt{737}}{32}\approx -0.317116998$$

Steps Using the Quadratic Formula

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

$$-16x^{2}+17x+7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 17 for b, and 7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-17±\sqrt{17^{2}-4\left(-16\right)\times 7}}{2\left(-16\right)}$$

Square 17.

$$x=\frac{-17±\sqrt{289-4\left(-16\right)\times 7}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$x=\frac{-17±\sqrt{289+64\times 7}}{2\left(-16\right)}$$

Multiply 64 times 7.

$$x=\frac{-17±\sqrt{289+448}}{2\left(-16\right)}$$

Add 289 to 448.

$$x=\frac{-17±\sqrt{737}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$x=\frac{-17±\sqrt{737}}{-32}$$

Now solve the equation x=\frac{-17±\sqrt{737}}{-32} when ± is plus. Add -17 to \sqrt{737}.

$$x=\frac{\sqrt{737}-17}{-32}$$

Divide -17+\sqrt{737} by -32.

$$x=\frac{17-\sqrt{737}}{32}$$

Now solve the equation x=\frac{-17±\sqrt{737}}{-32} when ± is minus. Subtract \sqrt{737} from -17.

$$x=\frac{-\sqrt{737}-17}{-32}$$

Divide -17-\sqrt{737} by -32.

$$x=\frac{\sqrt{737}+17}{32}$$

The equation is now solved.

$$x=\frac{17-\sqrt{737}}{32} x=\frac{\sqrt{737}+17}{32}$$

Steps for Completing the Square

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

$$-16x^{2}+17x+7=0$$

Subtract 7 from both sides. Anything subtracted from zero gives its negation.

$$-16x^{2}+17x=-7$$

Divide both sides by -16.

$$\frac{-16x^{2}+17x}{-16}=-\frac{7}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$x^{2}+\frac{17}{-16}x=-\frac{7}{-16}$$

Divide 17 by -16.

$$x^{2}-\frac{17}{16}x=-\frac{7}{-16}$$

Divide -7 by -16.

$$x^{2}-\frac{17}{16}x=\frac{7}{16}$$

Divide -\frac{17}{16}, the coefficient of the x term, by 2 to get -\frac{17}{32}. Then add the square of -\frac{17}{32} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{17}{16}x+\left(-\frac{17}{32}\right)^{2}=\frac{7}{16}+\left(-\frac{17}{32}\right)^{2}$$

Square -\frac{17}{32} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{17}{16}x+\frac{289}{1024}=\frac{7}{16}+\frac{289}{1024}$$

Add \frac{7}{16} to \frac{289}{1024} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{17}{16}x+\frac{289}{1024}=\frac{737}{1024}$$

Factor x^{2}-\frac{17}{16}x+\frac{289}{1024}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{17}{32}\right)^{2}=\frac{737}{1024}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{17}{32}\right)^{2}}=\sqrt{\frac{737}{1024}}$$

Simplify.

$$x-\frac{17}{32}=\frac{\sqrt{737}}{32} x-\frac{17}{32}=-\frac{\sqrt{737}}{32}$$

Add \frac{17}{32} to both sides of the equation.

$$x=\frac{\sqrt{737}+17}{32} x=\frac{17-\sqrt{737}}{32}$$