Solve 90(x-9)(x-10)=20(x-10)

$$90(x-9)(x-10)=20(x-10)$$

Solve for x

$$x=10$$

$$x = \frac{83}{9} = 9\frac{2}{9} \approx 9.222222222$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 90 by x-9.

$$\left(90x-810\right)\left(x-10\right)=20\left(x-10\right)$$

Use the distributive property to multiply 90x-810 by x-10 and combine like terms.

$$90x^{2}-1710x+8100=20\left(x-10\right)$$

Use the distributive property to multiply 20 by x-10.

$$90x^{2}-1710x+8100=20x-200$$

Subtract 20x from both sides.

$$90x^{2}-1710x+8100-20x=-200$$

Combine -1710x and -20x to get -1730x.

$$90x^{2}-1730x+8100=-200$$

Add 200 to both sides.

$$90x^{2}-1730x+8100+200=0$$

Add 8100 and 200 to get 8300.

$$90x^{2}-1730x+8300=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 90 for a, -1730 for b, and 8300 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1730\right)±\sqrt{\left(-1730\right)^{2}-4\times 90\times 8300}}{2\times 90}$$

Square -1730.

$$x=\frac{-\left(-1730\right)±\sqrt{2992900-4\times 90\times 8300}}{2\times 90}$$

Multiply -4 times 90.

$$x=\frac{-\left(-1730\right)±\sqrt{2992900-360\times 8300}}{2\times 90}$$

Multiply -360 times 8300.

$$x=\frac{-\left(-1730\right)±\sqrt{2992900-2988000}}{2\times 90}$$

Add 2992900 to -2988000.

$$x=\frac{-\left(-1730\right)±\sqrt{4900}}{2\times 90}$$

Take the square root of 4900.

$$x=\frac{-\left(-1730\right)±70}{2\times 90}$$

The opposite of -1730 is 1730.

$$x=\frac{1730±70}{2\times 90}$$

Multiply 2 times 90.

$$x=\frac{1730±70}{180}$$

Now solve the equation x=\frac{1730±70}{180} when ± is plus. Add 1730 to 70.

$$x=\frac{1800}{180}$$

Divide 1800 by 180.

$$x=10$$

Now solve the equation x=\frac{1730±70}{180} when ± is minus. Subtract 70 from 1730.

$$x=\frac{1660}{180}$$

Reduce the fraction \frac{1660}{180} to lowest terms by extracting and canceling out 20.

$$x=\frac{83}{9}$$

The equation is now solved.

$$x=10 x=\frac{83}{9}$$

Steps for Completing the Square

Use the distributive property to multiply 90 by x-9.

$$\left(90x-810\right)\left(x-10\right)=20\left(x-10\right)$$

Use the distributive property to multiply 90x-810 by x-10 and combine like terms.

$$90x^{2}-1710x+8100=20\left(x-10\right)$$

Use the distributive property to multiply 20 by x-10.

$$90x^{2}-1710x+8100=20x-200$$

Subtract 20x from both sides.

$$90x^{2}-1710x+8100-20x=-200$$

Combine -1710x and -20x to get -1730x.

$$90x^{2}-1730x+8100=-200$$

Subtract 8100 from both sides.

$$90x^{2}-1730x=-200-8100$$

Subtract 8100 from -200 to get -8300.

$$90x^{2}-1730x=-8300$$

Divide both sides by 90.

$$\frac{90x^{2}-1730x}{90}=-\frac{8300}{90}$$

Dividing by 90 undoes the multiplication by 90.

$$x^{2}+\left(-\frac{1730}{90}\right)x=-\frac{8300}{90}$$

Reduce the fraction \frac{-1730}{90} to lowest terms by extracting and canceling out 10.

$$x^{2}-\frac{173}{9}x=-\frac{8300}{90}$$

Reduce the fraction \frac{-8300}{90} to lowest terms by extracting and canceling out 10.

$$x^{2}-\frac{173}{9}x=-\frac{830}{9}$$

Divide -\frac{173}{9}, the coefficient of the x term, by 2 to get -\frac{173}{18}. Then add the square of -\frac{173}{18} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{173}{9}x+\left(-\frac{173}{18}\right)^{2}=-\frac{830}{9}+\left(-\frac{173}{18}\right)^{2}$$

Square -\frac{173}{18} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{173}{9}x+\frac{29929}{324}=-\frac{830}{9}+\frac{29929}{324}$$

Add -\frac{830}{9} to \frac{29929}{324} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{173}{9}x+\frac{29929}{324}=\frac{49}{324}$$

Factor x^{2}-\frac{173}{9}x+\frac{29929}{324}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{173}{18}\right)^{2}=\frac{49}{324}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{173}{18}\right)^{2}}=\sqrt{\frac{49}{324}}$$

Simplify.

$$x-\frac{173}{18}=\frac{7}{18} x-\frac{173}{18}=-\frac{7}{18}$$

Add \frac{173}{18} to both sides of the equation.

$$x=10 x=\frac{83}{9}$$