Solve 0=(44-25x+x^2

$$0 = ( 44 - 25 x + x ^ { 2 }$$

Solve for x

$$x = \frac{\sqrt{449} + 25}{2} \approx 23.09481005$$

$$x = \frac{25 - \sqrt{449}}{2} \approx 1.90518995$$

Steps Using the Quadratic Formula

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

$$44-25x+x^{2}=0$$

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.

$$x^{2}-25x+44=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -25 for b, and 44 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-25\right)±\sqrt{\left(-25\right)^{2}-4\times 44}}{2}$$

Square -25.

$$x=\frac{-\left(-25\right)±\sqrt{625-4\times 44}}{2}$$

Multiply -4 times 44.

$$x=\frac{-\left(-25\right)±\sqrt{625-176}}{2}$$

Add 625 to -176.

$$x=\frac{-\left(-25\right)±\sqrt{449}}{2}$$

The opposite of -25 is 25.

$$x=\frac{25±\sqrt{449}}{2}$$

Now solve the equation x=\frac{25±\sqrt{449}}{2} when ± is plus. Add 25 to \sqrt{449}.

$$x=\frac{\sqrt{449}+25}{2}$$

Now solve the equation x=\frac{25±\sqrt{449}}{2} when ± is minus. Subtract \sqrt{449} from 25.

$$x=\frac{25-\sqrt{449}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{449}+25}{2} x=\frac{25-\sqrt{449}}{2}$$

Steps for Completing the Square

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

$$44-25x+x^{2}=0$$

Subtract 44 from both sides. Anything subtracted from zero gives its negation.

$$-25x+x^{2}=-44$$

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.

$$x^{2}-25x=-44$$

Divide -25, the coefficient of the x term, by 2 to get -\frac{25}{2}. Then add the square of -\frac{25}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-25x+\left(-\frac{25}{2}\right)^{2}=-44+\left(-\frac{25}{2}\right)^{2}$$

Square -\frac{25}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-25x+\frac{625}{4}=-44+\frac{625}{4}$$

Add -44 to \frac{625}{4}.

$$x^{2}-25x+\frac{625}{4}=\frac{449}{4}$$

Factor x^{2}-25x+\frac{625}{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}.

$$\left(x-\frac{25}{2}\right)^{2}=\frac{449}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{25}{2}\right)^{2}}=\sqrt{\frac{449}{4}}$$

Simplify.

$$x-\frac{25}{2}=\frac{\sqrt{449}}{2} x-\frac{25}{2}=-\frac{\sqrt{449}}{2}$$

Add \frac{25}{2} to both sides of the equation.

$$x=\frac{\sqrt{449}+25}{2} x=\frac{25-\sqrt{449}}{2}$$