Solve x=-16x^2+75x+200

$$x = - 16 x ^ { 2 } + 75 x + 200$$

Solve for x

$$x = \frac{\sqrt{4569} + 37}{16} \approx 6.537148654$$

$$x=\frac{37-\sqrt{4569}}{16}\approx -1.912148654$$

Steps Using the Quadratic Formula

Add 16x^{2} to both sides.

$$x+16x^{2}=75x+200$$

Subtract 75x from both sides.

$$x+16x^{2}-75x=200$$

Combine x and -75x to get -74x.

$$-74x+16x^{2}=200$$

Subtract 200 from both sides.

$$-74x+16x^{2}-200=0$$

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.

$$16x^{2}-74x-200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, -74 for b, and -200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-74\right)±\sqrt{\left(-74\right)^{2}-4\times 16\left(-200\right)}}{2\times 16}$$

Square -74.

$$x=\frac{-\left(-74\right)±\sqrt{5476-4\times 16\left(-200\right)}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-\left(-74\right)±\sqrt{5476-64\left(-200\right)}}{2\times 16}$$

Multiply -64 times -200.

$$x=\frac{-\left(-74\right)±\sqrt{5476+12800}}{2\times 16}$$

Add 5476 to 12800.

$$x=\frac{-\left(-74\right)±\sqrt{18276}}{2\times 16}$$

Take the square root of 18276.

$$x=\frac{-\left(-74\right)±2\sqrt{4569}}{2\times 16}$$

The opposite of -74 is 74.

$$x=\frac{74±2\sqrt{4569}}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{74±2\sqrt{4569}}{32}$$

Now solve the equation x=\frac{74±2\sqrt{4569}}{32} when ± is plus. Add 74 to 2\sqrt{4569}.

$$x=\frac{2\sqrt{4569}+74}{32}$$

Divide 74+2\sqrt{4569} by 32.

$$x=\frac{\sqrt{4569}+37}{16}$$

Now solve the equation x=\frac{74±2\sqrt{4569}}{32} when ± is minus. Subtract 2\sqrt{4569} from 74.

$$x=\frac{74-2\sqrt{4569}}{32}$$

Divide 74-2\sqrt{4569} by 32.

$$x=\frac{37-\sqrt{4569}}{16}$$

The equation is now solved.

$$x=\frac{\sqrt{4569}+37}{16} x=\frac{37-\sqrt{4569}}{16}$$

Steps for Completing the Square

Add 16x^{2} to both sides.

$$x+16x^{2}=75x+200$$

Subtract 75x from both sides.

$$x+16x^{2}-75x=200$$

Combine x and -75x to get -74x.

$$-74x+16x^{2}=200$$

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.

$$16x^{2}-74x=200$$

Divide both sides by 16.

$$\frac{16x^{2}-74x}{16}=\frac{200}{16}$$

Dividing by 16 undoes the multiplication by 16.

$$x^{2}+\left(-\frac{74}{16}\right)x=\frac{200}{16}$$

Reduce the fraction \frac{-74}{16} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{37}{8}x=\frac{200}{16}$$

Reduce the fraction \frac{200}{16} to lowest terms by extracting and canceling out 8.

$$x^{2}-\frac{37}{8}x=\frac{25}{2}$$

Divide -\frac{37}{8}, the coefficient of the x term, by 2 to get -\frac{37}{16}. Then add the square of -\frac{37}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{37}{8}x+\left(-\frac{37}{16}\right)^{2}=\frac{25}{2}+\left(-\frac{37}{16}\right)^{2}$$

Square -\frac{37}{16} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{37}{8}x+\frac{1369}{256}=\frac{25}{2}+\frac{1369}{256}$$

Add \frac{25}{2} to \frac{1369}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{37}{8}x+\frac{1369}{256}=\frac{4569}{256}$$

Factor x^{2}-\frac{37}{8}x+\frac{1369}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{37}{16}\right)^{2}=\frac{4569}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{37}{16}\right)^{2}}=\sqrt{\frac{4569}{256}}$$

Simplify.

$$x-\frac{37}{16}=\frac{\sqrt{4569}}{16} x-\frac{37}{16}=-\frac{\sqrt{4569}}{16}$$

Add \frac{37}{16} to both sides of the equation.

$$x=\frac{\sqrt{4569}+37}{16} x=\frac{37-\sqrt{4569}}{16}$$