Solve x^2-226x+1105=0

$${ x }^{ 2 } -226x+1105=0$$

Solve for x

$$x=5$$

$$x=221$$

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.

$$x^{2}-226x+1105=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -226 for b, and 1105 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-226\right)±\sqrt{\left(-226\right)^{2}-4\times 1105}}{2}$$

Square -226.

$$x=\frac{-\left(-226\right)±\sqrt{51076-4\times 1105}}{2}$$

Multiply -4 times 1105.

$$x=\frac{-\left(-226\right)±\sqrt{51076-4420}}{2}$$

Add 51076 to -4420.

$$x=\frac{-\left(-226\right)±\sqrt{46656}}{2}$$

Take the square root of 46656.

$$x=\frac{-\left(-226\right)±216}{2}$$

The opposite of -226 is 226.

$$x=\frac{226±216}{2}$$

Now solve the equation x=\frac{226±216}{2} when ± is plus. Add 226 to 216.

$$x=\frac{442}{2}$$

Divide 442 by 2.

$$x=221$$

Now solve the equation x=\frac{226±216}{2} when ± is minus. Subtract 216 from 226.

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

Divide 10 by 2.

$$x=5$$

The equation is now solved.

$$x=221 x=5$$

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.

$$x^{2}-226x+1105=0$$

Subtract 1105 from both sides of the equation.

$$x^{2}-226x+1105-1105=-1105$$

Subtracting 1105 from itself leaves 0.

$$x^{2}-226x=-1105$$

Divide -226, the coefficient of the x term, by 2 to get -113. Then add the square of -113 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-226x+\left(-113\right)^{2}=-1105+\left(-113\right)^{2}$$

Square -113.

$$x^{2}-226x+12769=-1105+12769$$

Add -1105 to 12769.

$$x^{2}-226x+12769=11664$$

Factor x^{2}-226x+12769. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-113\right)^{2}=11664$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-113\right)^{2}}=\sqrt{11664}$$

Simplify.

$$x-113=108 x-113=-108$$

Add 113 to both sides of the equation.

$$x=221 x=5$$