Solve (133-x)^2=12x(133-x)

$$( 133 - x ) ^ { 2 } = 12 x ( 133 - x )$$

Solve for x

$$x=133$$

$$x = \frac{133}{13} = 10\frac{3}{13} \approx 10.230769231$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(133-x\right)^{2}.

$$17689-266x+x^{2}=12x\left(133-x\right)$$

Use the distributive property to multiply 12x by 133-x.

$$17689-266x+x^{2}=1596x-12x^{2}$$

Subtract 1596x from both sides.

$$17689-266x+x^{2}-1596x=-12x^{2}$$

Combine -266x and -1596x to get -1862x.

$$17689-1862x+x^{2}=-12x^{2}$$

Add 12x^{2} to both sides.

$$17689-1862x+x^{2}+12x^{2}=0$$

Combine x^{2} and 12x^{2} to get 13x^{2}.

$$17689-1862x+13x^{2}=0$$

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.

$$13x^{2}-1862x+17689=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 13 for a, -1862 for b, and 17689 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1862\right)±\sqrt{\left(-1862\right)^{2}-4\times 13\times 17689}}{2\times 13}$$

Square -1862.

$$x=\frac{-\left(-1862\right)±\sqrt{3467044-4\times 13\times 17689}}{2\times 13}$$

Multiply -4 times 13.

$$x=\frac{-\left(-1862\right)±\sqrt{3467044-52\times 17689}}{2\times 13}$$

Multiply -52 times 17689.

$$x=\frac{-\left(-1862\right)±\sqrt{3467044-919828}}{2\times 13}$$

Add 3467044 to -919828.

$$x=\frac{-\left(-1862\right)±\sqrt{2547216}}{2\times 13}$$

Take the square root of 2547216.

$$x=\frac{-\left(-1862\right)±1596}{2\times 13}$$

The opposite of -1862 is 1862.

$$x=\frac{1862±1596}{2\times 13}$$

Multiply 2 times 13.

$$x=\frac{1862±1596}{26}$$

Now solve the equation x=\frac{1862±1596}{26} when ± is plus. Add 1862 to 1596.

$$x=\frac{3458}{26}$$

Divide 3458 by 26.

$$x=133$$

Now solve the equation x=\frac{1862±1596}{26} when ± is minus. Subtract 1596 from 1862.

$$x=\frac{266}{26}$$

Reduce the fraction \frac{266}{26} to lowest terms by extracting and canceling out 2.

$$x=\frac{133}{13}$$

The equation is now solved.

$$x=133 x=\frac{133}{13}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(133-x\right)^{2}.

$$17689-266x+x^{2}=12x\left(133-x\right)$$

Use the distributive property to multiply 12x by 133-x.

$$17689-266x+x^{2}=1596x-12x^{2}$$

Subtract 1596x from both sides.

$$17689-266x+x^{2}-1596x=-12x^{2}$$

Combine -266x and -1596x to get -1862x.

$$17689-1862x+x^{2}=-12x^{2}$$

Add 12x^{2} to both sides.

$$17689-1862x+x^{2}+12x^{2}=0$$

Combine x^{2} and 12x^{2} to get 13x^{2}.

$$17689-1862x+13x^{2}=0$$

Subtract 17689 from both sides. Anything subtracted from zero gives its negation.

$$-1862x+13x^{2}=-17689$$

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.

$$13x^{2}-1862x=-17689$$

Divide both sides by 13.

$$\frac{13x^{2}-1862x}{13}=-\frac{17689}{13}$$

Dividing by 13 undoes the multiplication by 13.

$$x^{2}-\frac{1862}{13}x=-\frac{17689}{13}$$

Divide -\frac{1862}{13}, the coefficient of the x term, by 2 to get -\frac{931}{13}. Then add the square of -\frac{931}{13} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1862}{13}x+\left(-\frac{931}{13}\right)^{2}=-\frac{17689}{13}+\left(-\frac{931}{13}\right)^{2}$$

Square -\frac{931}{13} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1862}{13}x+\frac{866761}{169}=-\frac{17689}{13}+\frac{866761}{169}$$

Add -\frac{17689}{13} to \frac{866761}{169} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1862}{13}x+\frac{866761}{169}=\frac{636804}{169}$$

Factor x^{2}-\frac{1862}{13}x+\frac{866761}{169}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{931}{13}\right)^{2}=\frac{636804}{169}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{931}{13}\right)^{2}}=\sqrt{\frac{636804}{169}}$$

Simplify.

$$x-\frac{931}{13}=\frac{798}{13} x-\frac{931}{13}=-\frac{798}{13}$$

Add \frac{931}{13} to both sides of the equation.

$$x=133 x=\frac{133}{13}$$