Solve 16x^2-64x+64=12x

$$16 x ^ { 2 } - 64 x + 64 = 12 x$$

Solve for x

$$x = \frac{\sqrt{105} + 19}{8} \approx 3.655868846$$

$$x = \frac{19 - \sqrt{105}}{8} \approx 1.094131154$$

Steps Using the Quadratic Formula

Subtract 12x from both sides.

$$16x^{2}-64x+64-12x=0$$

Combine -64x and -12x to get -76x.

$$16x^{2}-76x+64=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, -76 for b, and 64 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-76\right)±\sqrt{\left(-76\right)^{2}-4\times 16\times 64}}{2\times 16}$$

Square -76.

$$x=\frac{-\left(-76\right)±\sqrt{5776-4\times 16\times 64}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-\left(-76\right)±\sqrt{5776-64\times 64}}{2\times 16}$$

Multiply -64 times 64.

$$x=\frac{-\left(-76\right)±\sqrt{5776-4096}}{2\times 16}$$

Add 5776 to -4096.

$$x=\frac{-\left(-76\right)±\sqrt{1680}}{2\times 16}$$

Take the square root of 1680.

$$x=\frac{-\left(-76\right)±4\sqrt{105}}{2\times 16}$$

The opposite of -76 is 76.

$$x=\frac{76±4\sqrt{105}}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{76±4\sqrt{105}}{32}$$

Now solve the equation x=\frac{76±4\sqrt{105}}{32} when ± is plus. Add 76 to 4\sqrt{105}.

$$x=\frac{4\sqrt{105}+76}{32}$$

Divide 76+4\sqrt{105} by 32.

$$x=\frac{\sqrt{105}+19}{8}$$

Now solve the equation x=\frac{76±4\sqrt{105}}{32} when ± is minus. Subtract 4\sqrt{105} from 76.

$$x=\frac{76-4\sqrt{105}}{32}$$

Divide 76-4\sqrt{105} by 32.

$$x=\frac{19-\sqrt{105}}{8}$$

The equation is now solved.

$$x=\frac{\sqrt{105}+19}{8} x=\frac{19-\sqrt{105}}{8}$$

Steps for Completing the Square

Subtract 12x from both sides.

$$16x^{2}-64x+64-12x=0$$

Combine -64x and -12x to get -76x.

$$16x^{2}-76x+64=0$$

Subtract 64 from both sides. Anything subtracted from zero gives its negation.

$$16x^{2}-76x=-64$$

Divide both sides by 16.

$$\frac{16x^{2}-76x}{16}=-\frac{64}{16}$$

Dividing by 16 undoes the multiplication by 16.

$$x^{2}+\left(-\frac{76}{16}\right)x=-\frac{64}{16}$$

Reduce the fraction \frac{-76}{16} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{19}{4}x=-\frac{64}{16}$$

Divide -64 by 16.

$$x^{2}-\frac{19}{4}x=-4$$

Divide -\frac{19}{4}, the coefficient of the x term, by 2 to get -\frac{19}{8}. Then add the square of -\frac{19}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{19}{4}x+\left(-\frac{19}{8}\right)^{2}=-4+\left(-\frac{19}{8}\right)^{2}$$

Square -\frac{19}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{19}{4}x+\frac{361}{64}=-4+\frac{361}{64}$$

Add -4 to \frac{361}{64}.

$$x^{2}-\frac{19}{4}x+\frac{361}{64}=\frac{105}{64}$$

Factor x^{2}-\frac{19}{4}x+\frac{361}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{19}{8}\right)^{2}=\frac{105}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{19}{8}\right)^{2}}=\sqrt{\frac{105}{64}}$$

Simplify.

$$x-\frac{19}{8}=\frac{\sqrt{105}}{8} x-\frac{19}{8}=-\frac{\sqrt{105}}{8}$$

Add \frac{19}{8} to both sides of the equation.

$$x=\frac{\sqrt{105}+19}{8} x=\frac{19-\sqrt{105}}{8}$$