Solve 4x+1/x-3=3x-8/x+1

$$\frac { 4 x + 1 } { x - 3 } = \frac { 3 x - 8 } { x + 1 }$$

Solve for x

$$x=-23$$

$$x=1$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -1,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x+1\right), the least common multiple of x-3,x+1.

$$\left(x+1\right)\left(4x+1\right)=\left(x-3\right)\left(3x-8\right)$$

Use the distributive property to multiply x+1 by 4x+1 and combine like terms.

$$4x^{2}+5x+1=\left(x-3\right)\left(3x-8\right)$$

Use the distributive property to multiply x-3 by 3x-8 and combine like terms.

$$4x^{2}+5x+1=3x^{2}-17x+24$$

Subtract 3x^{2} from both sides.

$$4x^{2}+5x+1-3x^{2}=-17x+24$$

Combine 4x^{2} and -3x^{2} to get x^{2}.

$$x^{2}+5x+1=-17x+24$$

Add 17x to both sides.

$$x^{2}+5x+1+17x=24$$

Combine 5x and 17x to get 22x.

$$x^{2}+22x+1=24$$

Subtract 24 from both sides.

$$x^{2}+22x+1-24=0$$

Subtract 24 from 1 to get -23.

$$x^{2}+22x-23=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 22 for b, and -23 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-22±\sqrt{22^{2}-4\left(-23\right)}}{2}$$

Square 22.

$$x=\frac{-22±\sqrt{484-4\left(-23\right)}}{2}$$

Multiply -4 times -23.

$$x=\frac{-22±\sqrt{484+92}}{2}$$

Add 484 to 92.

$$x=\frac{-22±\sqrt{576}}{2}$$

Take the square root of 576.

$$x=\frac{-22±24}{2}$$

Now solve the equation x=\frac{-22±24}{2} when ± is plus. Add -22 to 24.

$$x=\frac{2}{2}$$

Divide 2 by 2.

$$x=1$$

Now solve the equation x=\frac{-22±24}{2} when ± is minus. Subtract 24 from -22.

$$x=-\frac{46}{2}$$

Divide -46 by 2.

$$x=-23$$

The equation is now solved.

$$x=1 x=-23$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -1,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x+1\right), the least common multiple of x-3,x+1.

$$\left(x+1\right)\left(4x+1\right)=\left(x-3\right)\left(3x-8\right)$$

Use the distributive property to multiply x+1 by 4x+1 and combine like terms.

$$4x^{2}+5x+1=\left(x-3\right)\left(3x-8\right)$$

Use the distributive property to multiply x-3 by 3x-8 and combine like terms.

$$4x^{2}+5x+1=3x^{2}-17x+24$$

Subtract 3x^{2} from both sides.

$$4x^{2}+5x+1-3x^{2}=-17x+24$$

Combine 4x^{2} and -3x^{2} to get x^{2}.

$$x^{2}+5x+1=-17x+24$$

Add 17x to both sides.

$$x^{2}+5x+1+17x=24$$

Combine 5x and 17x to get 22x.

$$x^{2}+22x+1=24$$

Subtract 1 from both sides.

$$x^{2}+22x=24-1$$

Subtract 1 from 24 to get 23.

$$x^{2}+22x=23$$

Divide 22, the coefficient of the x term, by 2 to get 11. Then add the square of 11 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+22x+11^{2}=23+11^{2}$$

Square 11.

$$x^{2}+22x+121=23+121$$

Add 23 to 121.

$$x^{2}+22x+121=144$$

Factor x^{2}+22x+121. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+11\right)^{2}=144$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+11\right)^{2}}=\sqrt{144}$$

Simplify.

$$x+11=12 x+11=-12$$

Subtract 11 from both sides of the equation.

$$x=1 x=-23$$