Solve 12/13=frac{52^{2}x^2+33^2-25^2x^2}{104*33x}

$$\frac { 12 } { 13 } = \frac { 52 ^ { 2 } x ^ { 2 } + 33 ^ { 2 } - 25 ^ { 2 } x ^ { 2 } } { 104 \times 33 x }$$

Solve for x

$$x=1$$

$$x=\frac{11}{21}\approx 0.523809524$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 3432x, the least common multiple of 13,104\times 33x.

$$3168x=52^{2}x^{2}+33^{2}-25^{2}x^{2}$$

Calculate 52 to the power of 2 and get 2704.

$$3168x=2704x^{2}+33^{2}-25^{2}x^{2}$$

Calculate 33 to the power of 2 and get 1089.

$$3168x=2704x^{2}+1089-25^{2}x^{2}$$

Calculate 25 to the power of 2 and get 625.

$$3168x=2704x^{2}+1089-625x^{2}$$

Combine 2704x^{2} and -625x^{2} to get 2079x^{2}.

$$3168x=2079x^{2}+1089$$

Subtract 2079x^{2} from both sides.

$$3168x-2079x^{2}=1089$$

Subtract 1089 from both sides.

$$3168x-2079x^{2}-1089=0$$

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.

$$-2079x^{2}+3168x-1089=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2079 for a, 3168 for b, and -1089 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3168±\sqrt{3168^{2}-4\left(-2079\right)\left(-1089\right)}}{2\left(-2079\right)}$$

Square 3168.

$$x=\frac{-3168±\sqrt{10036224-4\left(-2079\right)\left(-1089\right)}}{2\left(-2079\right)}$$

Multiply -4 times -2079.

$$x=\frac{-3168±\sqrt{10036224+8316\left(-1089\right)}}{2\left(-2079\right)}$$

Multiply 8316 times -1089.

$$x=\frac{-3168±\sqrt{10036224-9056124}}{2\left(-2079\right)}$$

Add 10036224 to -9056124.

$$x=\frac{-3168±\sqrt{980100}}{2\left(-2079\right)}$$

Take the square root of 980100.

$$x=\frac{-3168±990}{2\left(-2079\right)}$$

Multiply 2 times -2079.

$$x=\frac{-3168±990}{-4158}$$

Now solve the equation x=\frac{-3168±990}{-4158} when ± is plus. Add -3168 to 990.

$$x=-\frac{2178}{-4158}$$

Reduce the fraction \frac{-2178}{-4158} to lowest terms by extracting and canceling out 198.

$$x=\frac{11}{21}$$

Now solve the equation x=\frac{-3168±990}{-4158} when ± is minus. Subtract 990 from -3168.

$$x=-\frac{4158}{-4158}$$

Divide -4158 by -4158.

$$x=1$$

The equation is now solved.

$$x=\frac{11}{21} x=1$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 3432x, the least common multiple of 13,104\times 33x.

$$3168x=52^{2}x^{2}+33^{2}-25^{2}x^{2}$$

Calculate 52 to the power of 2 and get 2704.

$$3168x=2704x^{2}+33^{2}-25^{2}x^{2}$$

Calculate 33 to the power of 2 and get 1089.

$$3168x=2704x^{2}+1089-25^{2}x^{2}$$

Calculate 25 to the power of 2 and get 625.

$$3168x=2704x^{2}+1089-625x^{2}$$

Combine 2704x^{2} and -625x^{2} to get 2079x^{2}.

$$3168x=2079x^{2}+1089$$

Subtract 2079x^{2} from both sides.

$$3168x-2079x^{2}=1089$$

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.

$$-2079x^{2}+3168x=1089$$

Divide both sides by -2079.

$$\frac{-2079x^{2}+3168x}{-2079}=\frac{1089}{-2079}$$

Dividing by -2079 undoes the multiplication by -2079.

$$x^{2}+\frac{3168}{-2079}x=\frac{1089}{-2079}$$

Reduce the fraction \frac{3168}{-2079} to lowest terms by extracting and canceling out 99.

$$x^{2}-\frac{32}{21}x=\frac{1089}{-2079}$$

Reduce the fraction \frac{1089}{-2079} to lowest terms by extracting and canceling out 99.

$$x^{2}-\frac{32}{21}x=-\frac{11}{21}$$

Divide -\frac{32}{21}, the coefficient of the x term, by 2 to get -\frac{16}{21}. Then add the square of -\frac{16}{21} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{32}{21}x+\left(-\frac{16}{21}\right)^{2}=-\frac{11}{21}+\left(-\frac{16}{21}\right)^{2}$$

Square -\frac{16}{21} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{32}{21}x+\frac{256}{441}=-\frac{11}{21}+\frac{256}{441}$$

Add -\frac{11}{21} to \frac{256}{441} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{32}{21}x+\frac{256}{441}=\frac{25}{441}$$

Factor x^{2}-\frac{32}{21}x+\frac{256}{441}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{16}{21}\right)^{2}=\frac{25}{441}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{16}{21}\right)^{2}}=\sqrt{\frac{25}{441}}$$

Simplify.

$$x-\frac{16}{21}=\frac{5}{21} x-\frac{16}{21}=-\frac{5}{21}$$

Add \frac{16}{21} to both sides of the equation.

$$x=1 x=\frac{11}{21}$$