Solve 0=-16x^2+67x+2

$$0 = -16 { x }^{ 2 } +67x+2$$

Solve for x

$$x = \frac{9 \sqrt{57} + 67}{32} \approx 4.217140935$$

$$x=\frac{67-9\sqrt{57}}{32}\approx -0.029640935$$

Steps Using the Quadratic Formula

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

$$-16x^{2}+67x+2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 67 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-67±\sqrt{67^{2}-4\left(-16\right)\times 2}}{2\left(-16\right)}$$

Square 67.

$$x=\frac{-67±\sqrt{4489-4\left(-16\right)\times 2}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$x=\frac{-67±\sqrt{4489+64\times 2}}{2\left(-16\right)}$$

Multiply 64 times 2.

$$x=\frac{-67±\sqrt{4489+128}}{2\left(-16\right)}$$

Add 4489 to 128.

$$x=\frac{-67±\sqrt{4617}}{2\left(-16\right)}$$

Take the square root of 4617.

$$x=\frac{-67±9\sqrt{57}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$x=\frac{-67±9\sqrt{57}}{-32}$$

Now solve the equation x=\frac{-67±9\sqrt{57}}{-32} when ± is plus. Add -67 to 9\sqrt{57}.

$$x=\frac{9\sqrt{57}-67}{-32}$$

Divide -67+9\sqrt{57} by -32.

$$x=\frac{67-9\sqrt{57}}{32}$$

Now solve the equation x=\frac{-67±9\sqrt{57}}{-32} when ± is minus. Subtract 9\sqrt{57} from -67.

$$x=\frac{-9\sqrt{57}-67}{-32}$$

Divide -67-9\sqrt{57} by -32.

$$x=\frac{9\sqrt{57}+67}{32}$$

The equation is now solved.

$$x=\frac{67-9\sqrt{57}}{32} x=\frac{9\sqrt{57}+67}{32}$$

Steps for Completing the Square

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

$$-16x^{2}+67x+2=0$$

Subtract 2 from both sides. Anything subtracted from zero gives its negation.

$$-16x^{2}+67x=-2$$

Divide both sides by -16.

$$\frac{-16x^{2}+67x}{-16}=-\frac{2}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$x^{2}+\frac{67}{-16}x=-\frac{2}{-16}$$

Divide 67 by -16.

$$x^{2}-\frac{67}{16}x=-\frac{2}{-16}$$

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

$$x^{2}-\frac{67}{16}x=\frac{1}{8}$$

Divide -\frac{67}{16}, the coefficient of the x term, by 2 to get -\frac{67}{32}. Then add the square of -\frac{67}{32} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{67}{16}x+\left(-\frac{67}{32}\right)^{2}=\frac{1}{8}+\left(-\frac{67}{32}\right)^{2}$$

Square -\frac{67}{32} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{67}{16}x+\frac{4489}{1024}=\frac{1}{8}+\frac{4489}{1024}$$

Add \frac{1}{8} to \frac{4489}{1024} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{67}{16}x+\frac{4489}{1024}=\frac{4617}{1024}$$

Factor x^{2}-\frac{67}{16}x+\frac{4489}{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}.

$$\left(x-\frac{67}{32}\right)^{2}=\frac{4617}{1024}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{67}{32}\right)^{2}}=\sqrt{\frac{4617}{1024}}$$

Simplify.

$$x-\frac{67}{32}=\frac{9\sqrt{57}}{32} x-\frac{67}{32}=-\frac{9\sqrt{57}}{32}$$

Add \frac{67}{32} to both sides of the equation.

$$x=\frac{9\sqrt{57}+67}{32} x=\frac{67-9\sqrt{57}}{32}$$