Solve 2x/x-2=5+13x^2/x-2

$$\frac { 2 x } { x - 2 } = 5 + \frac { 13 x ^ { 2 } } { x - 2 }$$

Solve for x

$$x=-1$$

$$x=\frac{10}{13}\approx 0.769230769$$

Steps Using Factoring By Grouping

Variable x cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by x-2.

$$2x=\left(x-2\right)\times 5+13x^{2}$$

Use the distributive property to multiply x-2 by 5.

$$2x=5x-10+13x^{2}$$

Subtract 5x from both sides.

$$2x-5x=-10+13x^{2}$$

Combine 2x and -5x to get -3x.

$$-3x=-10+13x^{2}$$

Subtract -10 from both sides.

$$-3x-\left(-10\right)=13x^{2}$$

The opposite of -10 is 10.

$$-3x+10=13x^{2}$$

Subtract 13x^{2} from both sides.

$$-3x+10-13x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-13x^{2}-3x+10=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -13x^{2}+ax+bx+10. To find a and b, set up a system to be solved.

$$a+b=-3 ab=-13\times 10=-130$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -130.

$$1,-130 2,-65 5,-26 10,-13$$

Calculate the sum for each pair.

$$1-130=-129 2-65=-63 5-26=-21 10-13=-3$$

The solution is the pair that gives sum -3.

$$a=10 b=-13$$

Rewrite -13x^{2}-3x+10 as \left(-13x^{2}+10x\right)+\left(-13x+10\right).

$$\left(-13x^{2}+10x\right)+\left(-13x+10\right)$$

Factor out -x in the first and -1 in the second group.

$$-x\left(13x-10\right)-\left(13x-10\right)$$

Factor out common term 13x-10 by using distributive property.

$$\left(13x-10\right)\left(-x-1\right)$$

To find equation solutions, solve 13x-10=0 and -x-1=0.

$$x=\frac{10}{13} x=-1$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by x-2.

$$2x=\left(x-2\right)\times 5+13x^{2}$$

Use the distributive property to multiply x-2 by 5.

$$2x=5x-10+13x^{2}$$

Subtract 5x from both sides.

$$2x-5x=-10+13x^{2}$$

Combine 2x and -5x to get -3x.

$$-3x=-10+13x^{2}$$

Subtract -10 from both sides.

$$-3x-\left(-10\right)=13x^{2}$$

The opposite of -10 is 10.

$$-3x+10=13x^{2}$$

Subtract 13x^{2} from both sides.

$$-3x+10-13x^{2}=0$$

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.

$$-13x^{2}-3x+10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -13 for a, -3 for b, and 10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-13\right)\times 10}}{2\left(-13\right)}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-13\right)\times 10}}{2\left(-13\right)}$$

Multiply -4 times -13.

$$x=\frac{-\left(-3\right)±\sqrt{9+52\times 10}}{2\left(-13\right)}$$

Multiply 52 times 10.

$$x=\frac{-\left(-3\right)±\sqrt{9+520}}{2\left(-13\right)}$$

Add 9 to 520.

$$x=\frac{-\left(-3\right)±\sqrt{529}}{2\left(-13\right)}$$

Take the square root of 529.

$$x=\frac{-\left(-3\right)±23}{2\left(-13\right)}$$

The opposite of -3 is 3.

$$x=\frac{3±23}{2\left(-13\right)}$$

Multiply 2 times -13.

$$x=\frac{3±23}{-26}$$

Now solve the equation x=\frac{3±23}{-26} when ± is plus. Add 3 to 23.

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

Divide 26 by -26.

$$x=-1$$

Now solve the equation x=\frac{3±23}{-26} when ± is minus. Subtract 23 from 3.

$$x=-\frac{20}{-26}$$

Reduce the fraction \frac{-20}{-26} to lowest terms by extracting and canceling out 2.

$$x=\frac{10}{13}$$

The equation is now solved.

$$x=-1 x=\frac{10}{13}$$

Steps for Completing the Square

Variable x cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by x-2.

$$2x=\left(x-2\right)\times 5+13x^{2}$$

Use the distributive property to multiply x-2 by 5.

$$2x=5x-10+13x^{2}$$

Subtract 5x from both sides.

$$2x-5x=-10+13x^{2}$$

Combine 2x and -5x to get -3x.

$$-3x=-10+13x^{2}$$

Subtract 13x^{2} from both sides.

$$-3x-13x^{2}=-10$$

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.

$$-13x^{2}-3x=-10$$

Divide both sides by -13.

$$\frac{-13x^{2}-3x}{-13}=-\frac{10}{-13}$$

Dividing by -13 undoes the multiplication by -13.

$$x^{2}+\left(-\frac{3}{-13}\right)x=-\frac{10}{-13}$$

Divide -3 by -13.

$$x^{2}+\frac{3}{13}x=-\frac{10}{-13}$$

Divide -10 by -13.

$$x^{2}+\frac{3}{13}x=\frac{10}{13}$$

Divide \frac{3}{13}, the coefficient of the x term, by 2 to get \frac{3}{26}. Then add the square of \frac{3}{26} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{3}{13}x+\left(\frac{3}{26}\right)^{2}=\frac{10}{13}+\left(\frac{3}{26}\right)^{2}$$

Square \frac{3}{26} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{3}{13}x+\frac{9}{676}=\frac{10}{13}+\frac{9}{676}$$

Add \frac{10}{13} to \frac{9}{676} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{3}{13}x+\frac{9}{676}=\frac{529}{676}$$

Factor x^{2}+\frac{3}{13}x+\frac{9}{676}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{26}\right)^{2}=\frac{529}{676}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{26}\right)^{2}}=\sqrt{\frac{529}{676}}$$

Simplify.

$$x+\frac{3}{26}=\frac{23}{26} x+\frac{3}{26}=-\frac{23}{26}$$

Subtract \frac{3}{26} from both sides of the equation.

$$x=\frac{10}{13} x=-1$$