Solve x=-16x^2+250

$$x=-16 { x }^{ 2 } +250$$

Solve for x

$$x = \frac{\sqrt{16001} - 1}{32} \approx 3.9217206$$

$$x=\frac{-\sqrt{16001}-1}{32}\approx -3.9842206$$

Steps Using the Quadratic Formula

Add 16x^{2} to both sides.

$$x+16x^{2}=250$$

Subtract 250 from both sides.

$$x+16x^{2}-250=0$$

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.

$$16x^{2}+x-250=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, 1 for b, and -250 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\times 16\left(-250\right)}}{2\times 16}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\times 16\left(-250\right)}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-1±\sqrt{1-64\left(-250\right)}}{2\times 16}$$

Multiply -64 times -250.

$$x=\frac{-1±\sqrt{1+16000}}{2\times 16}$$

Add 1 to 16000.

$$x=\frac{-1±\sqrt{16001}}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{-1±\sqrt{16001}}{32}$$

Now solve the equation x=\frac{-1±\sqrt{16001}}{32} when ± is plus. Add -1 to \sqrt{16001}.

$$x=\frac{\sqrt{16001}-1}{32}$$

Now solve the equation x=\frac{-1±\sqrt{16001}}{32} when ± is minus. Subtract \sqrt{16001} from -1.

$$x=\frac{-\sqrt{16001}-1}{32}$$

The equation is now solved.

$$x=\frac{\sqrt{16001}-1}{32} x=\frac{-\sqrt{16001}-1}{32}$$

Steps for Completing the Square

Add 16x^{2} to both sides.

$$x+16x^{2}=250$$

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.

$$16x^{2}+x=250$$

Divide both sides by 16.

$$\frac{16x^{2}+x}{16}=\frac{250}{16}$$

Dividing by 16 undoes the multiplication by 16.

$$x^{2}+\frac{1}{16}x=\frac{250}{16}$$

Reduce the fraction \frac{250}{16} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{1}{16}x=\frac{125}{8}$$

Divide \frac{1}{16}, the coefficient of the x term, by 2 to get \frac{1}{32}. Then add the square of \frac{1}{32} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{16}x+\left(\frac{1}{32}\right)^{2}=\frac{125}{8}+\left(\frac{1}{32}\right)^{2}$$

Square \frac{1}{32} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{16}x+\frac{1}{1024}=\frac{125}{8}+\frac{1}{1024}$$

Add \frac{125}{8} to \frac{1}{1024} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{16}x+\frac{1}{1024}=\frac{16001}{1024}$$

Factor x^{2}+\frac{1}{16}x+\frac{1}{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}.

$$\left(x+\frac{1}{32}\right)^{2}=\frac{16001}{1024}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{32}\right)^{2}}=\sqrt{\frac{16001}{1024}}$$

Simplify.

$$x+\frac{1}{32}=\frac{\sqrt{16001}}{32} x+\frac{1}{32}=-\frac{\sqrt{16001}}{32}$$

Subtract \frac{1}{32} from both sides of the equation.

$$x=\frac{\sqrt{16001}-1}{32} x=\frac{-\sqrt{16001}-1}{32}$$