Solve -16x^2+220x+138=883

$$-16 { x }^{ 2 } +220x+138 = 883$$

Solve for x

$$x = \frac{3 \sqrt{5} + 55}{8} \approx 7.713525492$$

$$x = \frac{55 - 3 \sqrt{5}}{8} \approx 6.036474508$$

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.

$$-16x^{2}+220x+138=883$$

Subtract 883 from both sides of the equation.

$$-16x^{2}+220x+138-883=883-883$$

Subtracting 883 from itself leaves 0.

$$-16x^{2}+220x+138-883=0$$

Subtract 883 from 138.

$$-16x^{2}+220x-745=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 220 for b, and -745 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-220±\sqrt{220^{2}-4\left(-16\right)\left(-745\right)}}{2\left(-16\right)}$$

Square 220.

$$x=\frac{-220±\sqrt{48400-4\left(-16\right)\left(-745\right)}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$x=\frac{-220±\sqrt{48400+64\left(-745\right)}}{2\left(-16\right)}$$

Multiply 64 times -745.

$$x=\frac{-220±\sqrt{48400-47680}}{2\left(-16\right)}$$

Add 48400 to -47680.

$$x=\frac{-220±\sqrt{720}}{2\left(-16\right)}$$

Take the square root of 720.

$$x=\frac{-220±12\sqrt{5}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$x=\frac{-220±12\sqrt{5}}{-32}$$

Now solve the equation x=\frac{-220±12\sqrt{5}}{-32} when ± is plus. Add -220 to 12\sqrt{5}.

$$x=\frac{12\sqrt{5}-220}{-32}$$

Divide -220+12\sqrt{5} by -32.

$$x=\frac{55-3\sqrt{5}}{8}$$

Now solve the equation x=\frac{-220±12\sqrt{5}}{-32} when ± is minus. Subtract 12\sqrt{5} from -220.

$$x=\frac{-12\sqrt{5}-220}{-32}$$

Divide -220-12\sqrt{5} by -32.

$$x=\frac{3\sqrt{5}+55}{8}$$

The equation is now solved.

$$x=\frac{55-3\sqrt{5}}{8} x=\frac{3\sqrt{5}+55}{8}$$

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.

$$-16x^{2}+220x+138=883$$

Subtract 138 from both sides of the equation.

$$-16x^{2}+220x+138-138=883-138$$

Subtracting 138 from itself leaves 0.

$$-16x^{2}+220x=883-138$$

Subtract 138 from 883.

$$-16x^{2}+220x=745$$

Divide both sides by -16.

$$\frac{-16x^{2}+220x}{-16}=\frac{745}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$x^{2}+\frac{220}{-16}x=\frac{745}{-16}$$

Reduce the fraction \frac{220}{-16} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{55}{4}x=\frac{745}{-16}$$

Divide 745 by -16.

$$x^{2}-\frac{55}{4}x=-\frac{745}{16}$$

Divide -\frac{55}{4}, the coefficient of the x term, by 2 to get -\frac{55}{8}. Then add the square of -\frac{55}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{55}{4}x+\left(-\frac{55}{8}\right)^{2}=-\frac{745}{16}+\left(-\frac{55}{8}\right)^{2}$$

Square -\frac{55}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{55}{4}x+\frac{3025}{64}=-\frac{745}{16}+\frac{3025}{64}$$

Add -\frac{745}{16} to \frac{3025}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{55}{4}x+\frac{3025}{64}=\frac{45}{64}$$

Factor x^{2}-\frac{55}{4}x+\frac{3025}{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}.

$$\left(x-\frac{55}{8}\right)^{2}=\frac{45}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{55}{8}\right)^{2}}=\sqrt{\frac{45}{64}}$$

Simplify.

$$x-\frac{55}{8}=\frac{3\sqrt{5}}{8} x-\frac{55}{8}=-\frac{3\sqrt{5}}{8}$$

Add \frac{55}{8} to both sides of the equation.

$$x=\frac{3\sqrt{5}+55}{8} x=\frac{55-3\sqrt{5}}{8}$$