Solve 90*(x-10)*(x-9)=1

$$90 \times (x-10) \times (x-9)=1$$

Solve for x

$$x=\frac{\sqrt{235}}{30}+\frac{19}{2}\approx 10.010990324$$

$$x=-\frac{\sqrt{235}}{30}+\frac{19}{2}\approx 8.989009676$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 90 by x-10.

$$\left(90x-900\right)\left(x-9\right)=1$$

Use the distributive property to multiply 90x-900 by x-9 and combine like terms.

$$90x^{2}-1710x+8100=1$$

Subtract 1 from both sides.

$$90x^{2}-1710x+8100-1=0$$

Subtract 1 from 8100 to get 8099.

$$90x^{2}-1710x+8099=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 90 for a, -1710 for b, and 8099 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1710\right)±\sqrt{\left(-1710\right)^{2}-4\times 90\times 8099}}{2\times 90}$$

Square -1710.

$$x=\frac{-\left(-1710\right)±\sqrt{2924100-4\times 90\times 8099}}{2\times 90}$$

Multiply -4 times 90.

$$x=\frac{-\left(-1710\right)±\sqrt{2924100-360\times 8099}}{2\times 90}$$

Multiply -360 times 8099.

$$x=\frac{-\left(-1710\right)±\sqrt{2924100-2915640}}{2\times 90}$$

Add 2924100 to -2915640.

$$x=\frac{-\left(-1710\right)±\sqrt{8460}}{2\times 90}$$

Take the square root of 8460.

$$x=\frac{-\left(-1710\right)±6\sqrt{235}}{2\times 90}$$

The opposite of -1710 is 1710.

$$x=\frac{1710±6\sqrt{235}}{2\times 90}$$

Multiply 2 times 90.

$$x=\frac{1710±6\sqrt{235}}{180}$$

Now solve the equation x=\frac{1710±6\sqrt{235}}{180} when ± is plus. Add 1710 to 6\sqrt{235}.

$$x=\frac{6\sqrt{235}+1710}{180}$$

Divide 1710+6\sqrt{235} by 180.

$$x=\frac{\sqrt{235}}{30}+\frac{19}{2}$$

Now solve the equation x=\frac{1710±6\sqrt{235}}{180} when ± is minus. Subtract 6\sqrt{235} from 1710.

$$x=\frac{1710-6\sqrt{235}}{180}$$

Divide 1710-6\sqrt{235} by 180.

$$x=-\frac{\sqrt{235}}{30}+\frac{19}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{235}}{30}+\frac{19}{2} x=-\frac{\sqrt{235}}{30}+\frac{19}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 90 by x-10.

$$\left(90x-900\right)\left(x-9\right)=1$$

Use the distributive property to multiply 90x-900 by x-9 and combine like terms.

$$90x^{2}-1710x+8100=1$$

Subtract 8100 from both sides.

$$90x^{2}-1710x=1-8100$$

Subtract 8100 from 1 to get -8099.

$$90x^{2}-1710x=-8099$$

Divide both sides by 90.

$$\frac{90x^{2}-1710x}{90}=-\frac{8099}{90}$$

Dividing by 90 undoes the multiplication by 90.

$$x^{2}+\left(-\frac{1710}{90}\right)x=-\frac{8099}{90}$$

Divide -1710 by 90.

$$x^{2}-19x=-\frac{8099}{90}$$

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.

$$x^{2}-19x+\left(-\frac{19}{2}\right)^{2}=-\frac{8099}{90}+\left(-\frac{19}{2}\right)^{2}$$

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

$$x^{2}-19x+\frac{361}{4}=-\frac{8099}{90}+\frac{361}{4}$$

Add -\frac{8099}{90} to \frac{361}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-19x+\frac{361}{4}=\frac{47}{180}$$

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}.

$$\left(x-\frac{19}{2}\right)^{2}=\frac{47}{180}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{19}{2}\right)^{2}}=\sqrt{\frac{47}{180}}$$

Simplify.

$$x-\frac{19}{2}=\frac{\sqrt{235}}{30} x-\frac{19}{2}=-\frac{\sqrt{235}}{30}$$

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

$$x=\frac{\sqrt{235}}{30}+\frac{19}{2} x=-\frac{\sqrt{235}}{30}+\frac{19}{2}$$