Solve 100-x^2=18^2-(16-x)

$$100- { x }^{ 2 } = { 18 }^{ 2 } -(16-x)$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{831}i-1}{2}\approx -0.5-14.413535305i$$

$$x=\frac{-1+\sqrt{831}i}{2}\approx -0.5+14.413535305i$$

Steps Using the Quadratic Formula

Calculate 18 to the power of 2 and get 324.

$$100-x^{2}=324-\left(16-x\right)$$

To find the opposite of 16-x, find the opposite of each term.

$$100-x^{2}=324-16+x$$

Subtract 16 from 324 to get 308.

$$100-x^{2}=308+x$$

Subtract 308 from both sides.

$$100-x^{2}-308=x$$

Subtract 308 from 100 to get -208.

$$-208-x^{2}=x$$

Subtract x from both sides.

$$-208-x^{2}-x=0$$

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.

$$-x^{2}-x-208=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -1 for b, and -208 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-1\right)\left(-208\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-1\right)±\sqrt{1+4\left(-208\right)}}{2\left(-1\right)}$$

Multiply 4 times -208.

$$x=\frac{-\left(-1\right)±\sqrt{1-832}}{2\left(-1\right)}$$

Add 1 to -832.

$$x=\frac{-\left(-1\right)±\sqrt{-831}}{2\left(-1\right)}$$

Take the square root of -831.

$$x=\frac{-\left(-1\right)±\sqrt{831}i}{2\left(-1\right)}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{831}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{1±\sqrt{831}i}{-2}$$

Now solve the equation x=\frac{1±\sqrt{831}i}{-2} when ± is plus. Add 1 to i\sqrt{831}.

$$x=\frac{1+\sqrt{831}i}{-2}$$

Divide 1+i\sqrt{831} by -2.

$$x=\frac{-\sqrt{831}i-1}{2}$$

Now solve the equation x=\frac{1±\sqrt{831}i}{-2} when ± is minus. Subtract i\sqrt{831} from 1.

$$x=\frac{-\sqrt{831}i+1}{-2}$$

Divide 1-i\sqrt{831} by -2.

$$x=\frac{-1+\sqrt{831}i}{2}$$

The equation is now solved.

$$x=\frac{-\sqrt{831}i-1}{2} x=\frac{-1+\sqrt{831}i}{2}$$

Steps for Completing the Square

Calculate 18 to the power of 2 and get 324.

$$100-x^{2}=324-\left(16-x\right)$$

To find the opposite of 16-x, find the opposite of each term.

$$100-x^{2}=324-16+x$$

Subtract 16 from 324 to get 308.

$$100-x^{2}=308+x$$

Subtract x from both sides.

$$100-x^{2}-x=308$$

Subtract 100 from both sides.

$$-x^{2}-x=308-100$$

Subtract 100 from 308 to get 208.

$$-x^{2}-x=208$$

Divide both sides by -1.

$$\frac{-x^{2}-x}{-1}=\frac{208}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{1}{-1}\right)x=\frac{208}{-1}$$

Divide -1 by -1.

$$x^{2}+x=\frac{208}{-1}$$

Divide 208 by -1.

$$x^{2}+x=-208$$

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

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=-208+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+x+\frac{1}{4}=-208+\frac{1}{4}$$

Add -208 to \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=-\frac{831}{4}$$

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

$$\left(x+\frac{1}{2}\right)^{2}=-\frac{831}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{-\frac{831}{4}}$$

Simplify.

$$x+\frac{1}{2}=\frac{\sqrt{831}i}{2} x+\frac{1}{2}=-\frac{\sqrt{831}i}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=\frac{-1+\sqrt{831}i}{2} x=\frac{-\sqrt{831}i-1}{2}$$