Solve 5x+7/x-1=3x+2

$$\frac { 5 x + 7 } { x - 1 } = 3 x + 2$$

Solve for x

$$x=-1$$

$$x=3$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by x-1.

$$5x+7=3x\left(x-1\right)+\left(x-1\right)\times 2$$

Use the distributive property to multiply 3x by x-1.

$$5x+7=3x^{2}-3x+\left(x-1\right)\times 2$$

Use the distributive property to multiply x-1 by 2.

$$5x+7=3x^{2}-3x+2x-2$$

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

$$5x+7=3x^{2}-x-2$$

Subtract 3x^{2} from both sides.

$$5x+7-3x^{2}=-x-2$$

Add x to both sides.

$$5x+7-3x^{2}+x=-2$$

Combine 5x and x to get 6x.

$$6x+7-3x^{2}=-2$$

Add 2 to both sides.

$$6x+7-3x^{2}+2=0$$

Add 7 and 2 to get 9.

$$6x+9-3x^{2}=0$$

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.

$$-3x^{2}+6x+9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 6 for b, and 9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-3\right)\times 9}}{2\left(-3\right)}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\left(-3\right)\times 9}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-6±\sqrt{36+12\times 9}}{2\left(-3\right)}$$

Multiply 12 times 9.

$$x=\frac{-6±\sqrt{36+108}}{2\left(-3\right)}$$

Add 36 to 108.

$$x=\frac{-6±\sqrt{144}}{2\left(-3\right)}$$

Take the square root of 144.

$$x=\frac{-6±12}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-6±12}{-6}$$

Now solve the equation x=\frac{-6±12}{-6} when ± is plus. Add -6 to 12.

$$x=\frac{6}{-6}$$

Divide 6 by -6.

$$x=-1$$

Now solve the equation x=\frac{-6±12}{-6} when ± is minus. Subtract 12 from -6.

$$x=-\frac{18}{-6}$$

Divide -18 by -6.

$$x=3$$

The equation is now solved.

$$x=-1 x=3$$

Steps for Completing the Square

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by x-1.

$$5x+7=3x\left(x-1\right)+\left(x-1\right)\times 2$$

Use the distributive property to multiply 3x by x-1.

$$5x+7=3x^{2}-3x+\left(x-1\right)\times 2$$

Use the distributive property to multiply x-1 by 2.

$$5x+7=3x^{2}-3x+2x-2$$

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

$$5x+7=3x^{2}-x-2$$

Subtract 3x^{2} from both sides.

$$5x+7-3x^{2}=-x-2$$

Add x to both sides.

$$5x+7-3x^{2}+x=-2$$

Combine 5x and x to get 6x.

$$6x+7-3x^{2}=-2$$

Subtract 7 from both sides.

$$6x-3x^{2}=-2-7$$

Subtract 7 from -2 to get -9.

$$6x-3x^{2}=-9$$

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.

$$-3x^{2}+6x=-9$$

Divide both sides by -3.

$$\frac{-3x^{2}+6x}{-3}=-\frac{9}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{6}{-3}x=-\frac{9}{-3}$$

Divide 6 by -3.

$$x^{2}-2x=-\frac{9}{-3}$$

Divide -9 by -3.

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

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=3+1$$

Add 3 to 1.

$$x^{2}-2x+1=4$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=4$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{4}$$

Simplify.

$$x-1=2 x-1=-2$$

Add 1 to both sides of the equation.

$$x=3 x=-1$$