Solve x^2-230x+1125=0

$${ x }^{ 2 } -230x+1125 = 0$$

Solve for x

$$x=5$$

$$x=225$$

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.

$$x^{2}-230x+1125=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -230 for b, and 1125 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-230\right)±\sqrt{\left(-230\right)^{2}-4\times 1125}}{2}$$

Square -230.

$$x=\frac{-\left(-230\right)±\sqrt{52900-4\times 1125}}{2}$$

Multiply -4 times 1125.

$$x=\frac{-\left(-230\right)±\sqrt{52900-4500}}{2}$$

Add 52900 to -4500.

$$x=\frac{-\left(-230\right)±\sqrt{48400}}{2}$$

Take the square root of 48400.

$$x=\frac{-\left(-230\right)±220}{2}$$

The opposite of -230 is 230.

$$x=\frac{230±220}{2}$$

Now solve the equation x=\frac{230±220}{2} when ± is plus. Add 230 to 220.

$$x=\frac{450}{2}$$

Divide 450 by 2.

$$x=225$$

Now solve the equation x=\frac{230±220}{2} when ± is minus. Subtract 220 from 230.

$$x=\frac{10}{2}$$

Divide 10 by 2.

$$x=5$$

The equation is now solved.

$$x=225 x=5$$

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.

$$x^{2}-230x+1125=0$$

Subtract 1125 from both sides of the equation.

$$x^{2}-230x+1125-1125=-1125$$

Subtracting 1125 from itself leaves 0.

$$x^{2}-230x=-1125$$

Divide -230, the coefficient of the x term, by 2 to get -115. Then add the square of -115 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-230x+\left(-115\right)^{2}=-1125+\left(-115\right)^{2}$$

Square -115.

$$x^{2}-230x+13225=-1125+13225$$

Add -1125 to 13225.

$$x^{2}-230x+13225=12100$$

Factor x^{2}-230x+13225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-115\right)^{2}=12100$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-115\right)^{2}}=\sqrt{12100}$$

Simplify.

$$x-115=110 x-115=-110$$

Add 115 to both sides of the equation.

$$x=225 x=5$$