Solve (x)=(x-9)^2

$$( x ) = ( x - 9 ) ^ { 2 }$$

Solve for x

$$x = \frac{\sqrt{37} + 19}{2} \approx 12.541381265$$

$$x = \frac{19 - \sqrt{37}}{2} \approx 6.458618735$$

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}.

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

Subtract x^{2} from both sides.

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

Add 18x to both sides.

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

Combine x and 18x to get 19x.

$$19x-x^{2}=81$$

Subtract 81 from both sides.

$$19x-x^{2}-81=0$$

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.

$$-x^{2}+19x-81=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 19 for b, and -81 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-19±\sqrt{19^{2}-4\left(-1\right)\left(-81\right)}}{2\left(-1\right)}$$

Square 19.

$$x=\frac{-19±\sqrt{361-4\left(-1\right)\left(-81\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-19±\sqrt{361+4\left(-81\right)}}{2\left(-1\right)}$$

Multiply 4 times -81.

$$x=\frac{-19±\sqrt{361-324}}{2\left(-1\right)}$$

Add 361 to -324.

$$x=\frac{-19±\sqrt{37}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-19±\sqrt{37}}{-2}$$

Now solve the equation x=\frac{-19±\sqrt{37}}{-2} when ± is plus. Add -19 to \sqrt{37}.

$$x=\frac{\sqrt{37}-19}{-2}$$

Divide -19+\sqrt{37} by -2.

$$x=\frac{19-\sqrt{37}}{2}$$

Now solve the equation x=\frac{-19±\sqrt{37}}{-2} when ± is minus. Subtract \sqrt{37} from -19.

$$x=\frac{-\sqrt{37}-19}{-2}$$

Divide -19-\sqrt{37} by -2.

$$x=\frac{\sqrt{37}+19}{2}$$

The equation is now solved.

$$x=\frac{19-\sqrt{37}}{2} x=\frac{\sqrt{37}+19}{2}$$

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}.

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

Subtract x^{2} from both sides.

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

Add 18x to both sides.

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

Combine x and 18x to get 19x.

$$19x-x^{2}=81$$

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.

$$-x^{2}+19x=81$$

Divide both sides by -1.

$$\frac{-x^{2}+19x}{-1}=\frac{81}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{19}{-1}x=\frac{81}{-1}$$

Divide 19 by -1.

$$x^{2}-19x=\frac{81}{-1}$$

Divide 81 by -1.

$$x^{2}-19x=-81$$

Divide -19, the coefficient of the x term, by 2 to get -\frac{19}{2}. Then add the square of -\frac{19}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-19x+\left(-\frac{19}{2}\right)^{2}=-81+\left(-\frac{19}{2}\right)^{2}$$

Square -\frac{19}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-19x+\frac{361}{4}=-81+\frac{361}{4}$$

Add -81 to \frac{361}{4}.

$$x^{2}-19x+\frac{361}{4}=\frac{37}{4}$$

Factor x^{2}-19x+\frac{361}{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}.

$$\left(x-\frac{19}{2}\right)^{2}=\frac{37}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{19}{2}\right)^{2}}=\sqrt{\frac{37}{4}}$$

Simplify.

$$x-\frac{19}{2}=\frac{\sqrt{37}}{2} x-\frac{19}{2}=-\frac{\sqrt{37}}{2}$$

Add \frac{19}{2} to both sides of the equation.

$$x=\frac{\sqrt{37}+19}{2} x=\frac{19-\sqrt{37}}{2}$$