Solve x^2-290x+18000=0

$${ x }^{ 2 } -290x+18000=0$$

Solve for x

$$x=90$$

$$x=200$$

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.

$$x^{2}-290x+18000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -290 for b, and 18000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-290\right)±\sqrt{\left(-290\right)^{2}-4\times 18000}}{2}$$

Square -290.

$$x=\frac{-\left(-290\right)±\sqrt{84100-4\times 18000}}{2}$$

Multiply -4 times 18000.

$$x=\frac{-\left(-290\right)±\sqrt{84100-72000}}{2}$$

Add 84100 to -72000.

$$x=\frac{-\left(-290\right)±\sqrt{12100}}{2}$$

Take the square root of 12100.

$$x=\frac{-\left(-290\right)±110}{2}$$

The opposite of -290 is 290.

$$x=\frac{290±110}{2}$$

Now solve the equation x=\frac{290±110}{2} when ± is plus. Add 290 to 110.

$$x=\frac{400}{2}$$

Divide 400 by 2.

$$x=200$$

Now solve the equation x=\frac{290±110}{2} when ± is minus. Subtract 110 from 290.

$$x=\frac{180}{2}$$

Divide 180 by 2.

$$x=90$$

The equation is now solved.

$$x=200 x=90$$

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.

$$x^{2}-290x+18000=0$$

Subtract 18000 from both sides of the equation.

$$x^{2}-290x+18000-18000=-18000$$

Subtracting 18000 from itself leaves 0.

$$x^{2}-290x=-18000$$

Divide -290, the coefficient of the x term, by 2 to get -145. Then add the square of -145 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-290x+\left(-145\right)^{2}=-18000+\left(-145\right)^{2}$$

Square -145.

$$x^{2}-290x+21025=-18000+21025$$

Add -18000 to 21025.

$$x^{2}-290x+21025=3025$$

Factor x^{2}-290x+21025. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-145\right)^{2}=3025$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-145\right)^{2}}=\sqrt{3025}$$

Simplify.

$$x-145=55 x-145=-55$$

Add 145 to both sides of the equation.

$$x=200 x=90$$