Solve 16x^2-64x-89=0

$$16 { x }^{ 2 } -64x-89=0$$

Solve for x

$$x=\frac{3\sqrt{17}}{4}+2\approx 5.092329219$$

$$x=-\frac{3\sqrt{17}}{4}+2\approx -1.092329219$$

Steps Using the Quadratic Formula

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$16x^{2}-64x-89=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, -64 for b, and -89 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-64\right)±\sqrt{\left(-64\right)^{2}-4\times 16\left(-89\right)}}{2\times 16}$$

Square -64.

$$x=\frac{-\left(-64\right)±\sqrt{4096-4\times 16\left(-89\right)}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-\left(-64\right)±\sqrt{4096-64\left(-89\right)}}{2\times 16}$$

Multiply -64 times -89.

$$x=\frac{-\left(-64\right)±\sqrt{4096+5696}}{2\times 16}$$

Add 4096 to 5696.

$$x=\frac{-\left(-64\right)±\sqrt{9792}}{2\times 16}$$

Take the square root of 9792.

$$x=\frac{-\left(-64\right)±24\sqrt{17}}{2\times 16}$$

The opposite of -64 is 64.

$$x=\frac{64±24\sqrt{17}}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{64±24\sqrt{17}}{32}$$

Now solve the equation x=\frac{64±24\sqrt{17}}{32} when ± is plus. Add 64 to 24\sqrt{17}.

$$x=\frac{24\sqrt{17}+64}{32}$$

Divide 64+24\sqrt{17} by 32.

$$x=\frac{3\sqrt{17}}{4}+2$$

Now solve the equation x=\frac{64±24\sqrt{17}}{32} when ± is minus. Subtract 24\sqrt{17} from 64.

$$x=\frac{64-24\sqrt{17}}{32}$$

Divide 64-24\sqrt{17} by 32.

$$x=-\frac{3\sqrt{17}}{4}+2$$

The equation is now solved.

$$x=\frac{3\sqrt{17}}{4}+2 x=-\frac{3\sqrt{17}}{4}+2$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$16x^{2}-64x-89=0$$

Add 89 to both sides of the equation.

$$16x^{2}-64x-89-\left(-89\right)=-\left(-89\right)$$

Subtracting -89 from itself leaves 0.

$$16x^{2}-64x=-\left(-89\right)$$

Subtract -89 from 0.

$$16x^{2}-64x=89$$

Divide both sides by 16.

$$\frac{16x^{2}-64x}{16}=\frac{89}{16}$$

Dividing by 16 undoes the multiplication by 16.

$$x^{2}+\left(-\frac{64}{16}\right)x=\frac{89}{16}$$

Divide -64 by 16.

$$x^{2}-4x=\frac{89}{16}$$

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

$$x^{2}-4x+\left(-2\right)^{2}=\frac{89}{16}+\left(-2\right)^{2}$$

Square -2.

$$x^{2}-4x+4=\frac{89}{16}+4$$

Add \frac{89}{16} to 4.

$$x^{2}-4x+4=\frac{153}{16}$$

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

$$\left(x-2\right)^{2}=\frac{153}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{\frac{153}{16}}$$

Simplify.

$$x-2=\frac{3\sqrt{17}}{4} x-2=-\frac{3\sqrt{17}}{4}$$

Add 2 to both sides of the equation.

$$x=\frac{3\sqrt{17}}{4}+2 x=-\frac{3\sqrt{17}}{4}+2$$