Solve 90000=120-6.25(x-9)^2

$$90000 = 120 - 6.25 ( x - 9 ) ^ { 2 }$$

Solve for x (complex solution)

$$x=-\frac{i\times 4\sqrt{22470}}{5}+9\approx 9-119.919973316i$$

$$x=\frac{i\times 4\sqrt{22470}}{5}+9\approx 9+119.919973316i$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-9\right)^{2}.

$$90000=120-6.25\left(x^{2}-18x+81\right)$$

Use the distributive property to multiply -6.25 by x^{2}-18x+81.

$$90000=120-6.25x^{2}+112.5x-506.25$$

Subtract 506.25 from 120 to get -386.25.

$$90000=-386.25-6.25x^{2}+112.5x$$

Swap sides so that all variable terms are on the left hand side.

$$-386.25-6.25x^{2}+112.5x=90000$$

Subtract 90000 from both sides.

$$-386.25-6.25x^{2}+112.5x-90000=0$$

Subtract 90000 from -386.25 to get -90386.25.

$$-90386.25-6.25x^{2}+112.5x=0$$

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.

$$-6.25x^{2}+112.5x-90386.25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6.25 for a, 112.5 for b, and -90386.25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-112.5±\sqrt{112.5^{2}-4\left(-6.25\right)\left(-90386.25\right)}}{2\left(-6.25\right)}$$

Square 112.5 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-112.5±\sqrt{12656.25-4\left(-6.25\right)\left(-90386.25\right)}}{2\left(-6.25\right)}$$

Multiply -4 times -6.25.

$$x=\frac{-112.5±\sqrt{12656.25+25\left(-90386.25\right)}}{2\left(-6.25\right)}$$

Multiply 25 times -90386.25.

$$x=\frac{-112.5±\sqrt{\frac{50625-9038625}{4}}}{2\left(-6.25\right)}$$

Add 12656.25 to -2259656.25 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-112.5±\sqrt{-2247000}}{2\left(-6.25\right)}$$

Take the square root of -2247000.

$$x=\frac{-112.5±10\sqrt{22470}i}{2\left(-6.25\right)}$$

Multiply 2 times -6.25.

$$x=\frac{-112.5±10\sqrt{22470}i}{-12.5}$$

Now solve the equation x=\frac{-112.5±10\sqrt{22470}i}{-12.5} when ± is plus. Add -112.5 to 10i\sqrt{22470}.

$$x=\frac{-112.5+10\sqrt{22470}i}{-12.5}$$

Divide -112.5+10i\sqrt{22470} by -12.5 by multiplying -112.5+10i\sqrt{22470} by the reciprocal of -12.5.

$$x=-\frac{4\sqrt{22470}i}{5}+9$$

Now solve the equation x=\frac{-112.5±10\sqrt{22470}i}{-12.5} when ± is minus. Subtract 10i\sqrt{22470} from -112.5.

$$x=\frac{-10\sqrt{22470}i-112.5}{-12.5}$$

Divide -112.5-10i\sqrt{22470} by -12.5 by multiplying -112.5-10i\sqrt{22470} by the reciprocal of -12.5.

$$x=\frac{4\sqrt{22470}i}{5}+9$$

The equation is now solved.

$$x=-\frac{4\sqrt{22470}i}{5}+9 x=\frac{4\sqrt{22470}i}{5}+9$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-9\right)^{2}.

$$90000=120-6.25\left(x^{2}-18x+81\right)$$

Use the distributive property to multiply -6.25 by x^{2}-18x+81.

$$90000=120-6.25x^{2}+112.5x-506.25$$

Subtract 506.25 from 120 to get -386.25.

$$90000=-386.25-6.25x^{2}+112.5x$$

Swap sides so that all variable terms are on the left hand side.

$$-386.25-6.25x^{2}+112.5x=90000$$

Add 386.25 to both sides.

$$-6.25x^{2}+112.5x=90000+386.25$$

Add 90000 and 386.25 to get 90386.25.

$$-6.25x^{2}+112.5x=90386.25$$

Divide both sides of the equation by -6.25, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-6.25x^{2}+112.5x}{-6.25}=\frac{90386.25}{-6.25}$$

Dividing by -6.25 undoes the multiplication by -6.25.

$$x^{2}+\frac{112.5}{-6.25}x=\frac{90386.25}{-6.25}$$

Divide 112.5 by -6.25 by multiplying 112.5 by the reciprocal of -6.25.

$$x^{2}-18x=\frac{90386.25}{-6.25}$$

Divide 90386.25 by -6.25 by multiplying 90386.25 by the reciprocal of -6.25.

$$x^{2}-18x=-14461.8$$

Divide -18, the coefficient of the x term, by 2 to get -9. Then add the square of -9 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-18x+\left(-9\right)^{2}=-14461.8+\left(-9\right)^{2}$$

Square -9.

$$x^{2}-18x+81=-14461.8+81$$

Add -14461.8 to 81.

$$x^{2}-18x+81=-14380.8$$

Factor x^{2}-18x+81. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-9\right)^{2}=-14380.8$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-9\right)^{2}}=\sqrt{-14380.8}$$

Simplify.

$$x-9=\frac{4\sqrt{22470}i}{5} x-9=-\frac{4\sqrt{22470}i}{5}$$

Add 9 to both sides of the equation.

$$x=\frac{4\sqrt{22470}i}{5}+9 x=-\frac{4\sqrt{22470}i}{5}+9$$