Solve 4x-8/x+5-3=x-8/2

$$\frac { 4 x - 8 } { x + 5 } - 3 = \frac { x - 8 } { 2 }$$

Solve for x

$$x=2$$

$$x=3$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -5 since division by zero is not defined. Multiply both sides of the equation by 2\left(x+5\right), the least common multiple of x+5,2.

$$2\left(4x-8\right)+2\left(x+5\right)\left(-3\right)=\left(x+5\right)\left(x-8\right)$$

Use the distributive property to multiply 2 by 4x-8.

$$8x-16+2\left(x+5\right)\left(-3\right)=\left(x+5\right)\left(x-8\right)$$

Multiply 2 and -3 to get -6.

$$8x-16-6\left(x+5\right)=\left(x+5\right)\left(x-8\right)$$

Use the distributive property to multiply -6 by x+5.

$$8x-16-6x-30=\left(x+5\right)\left(x-8\right)$$

Combine 8x and -6x to get 2x.

$$2x-16-30=\left(x+5\right)\left(x-8\right)$$

Subtract 30 from -16 to get -46.

$$2x-46=\left(x+5\right)\left(x-8\right)$$

Use the distributive property to multiply x+5 by x-8 and combine like terms.

$$2x-46=x^{2}-3x-40$$

Subtract x^{2} from both sides.

$$2x-46-x^{2}=-3x-40$$

Add 3x to both sides.

$$2x-46-x^{2}+3x=-40$$

Combine 2x and 3x to get 5x.

$$5x-46-x^{2}=-40$$

Add 40 to both sides.

$$5x-46-x^{2}+40=0$$

Add -46 and 40 to get -6.

$$5x-6-x^{2}=0$$

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.

$$-x^{2}+5x-6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 5 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\left(-1\right)\left(-6\right)}}{2\left(-1\right)}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\left(-1\right)\left(-6\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-5±\sqrt{25+4\left(-6\right)}}{2\left(-1\right)}$$

Multiply 4 times -6.

$$x=\frac{-5±\sqrt{25-24}}{2\left(-1\right)}$$

Add 25 to -24.

$$x=\frac{-5±\sqrt{1}}{2\left(-1\right)}$$

Take the square root of 1.

$$x=\frac{-5±1}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-5±1}{-2}$$

Now solve the equation x=\frac{-5±1}{-2} when ± is plus. Add -5 to 1.

$$x=-\frac{4}{-2}$$

Divide -4 by -2.

$$x=2$$

Now solve the equation x=\frac{-5±1}{-2} when ± is minus. Subtract 1 from -5.

$$x=-\frac{6}{-2}$$

Divide -6 by -2.

$$x=3$$

The equation is now solved.

$$x=2 x=3$$

Steps for Completing the Square

Variable x cannot be equal to -5 since division by zero is not defined. Multiply both sides of the equation by 2\left(x+5\right), the least common multiple of x+5,2.

$$2\left(4x-8\right)+2\left(x+5\right)\left(-3\right)=\left(x+5\right)\left(x-8\right)$$

Use the distributive property to multiply 2 by 4x-8.

$$8x-16+2\left(x+5\right)\left(-3\right)=\left(x+5\right)\left(x-8\right)$$

Multiply 2 and -3 to get -6.

$$8x-16-6\left(x+5\right)=\left(x+5\right)\left(x-8\right)$$

Use the distributive property to multiply -6 by x+5.

$$8x-16-6x-30=\left(x+5\right)\left(x-8\right)$$

Combine 8x and -6x to get 2x.

$$2x-16-30=\left(x+5\right)\left(x-8\right)$$

Subtract 30 from -16 to get -46.

$$2x-46=\left(x+5\right)\left(x-8\right)$$

Use the distributive property to multiply x+5 by x-8 and combine like terms.

$$2x-46=x^{2}-3x-40$$

Subtract x^{2} from both sides.

$$2x-46-x^{2}=-3x-40$$

Add 3x to both sides.

$$2x-46-x^{2}+3x=-40$$

Combine 2x and 3x to get 5x.

$$5x-46-x^{2}=-40$$

Add 46 to both sides.

$$5x-x^{2}=-40+46$$

Add -40 and 46 to get 6.

$$5x-x^{2}=6$$

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.

$$-x^{2}+5x=6$$

Divide both sides by -1.

$$\frac{-x^{2}+5x}{-1}=\frac{6}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{5}{-1}x=\frac{6}{-1}$$

Divide 5 by -1.

$$x^{2}-5x=\frac{6}{-1}$$

Divide 6 by -1.

$$x^{2}-5x=-6$$

Divide -5, the coefficient of the x term, by 2 to get -\frac{5}{2}. Then add the square of -\frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=-6+\left(-\frac{5}{2}\right)^{2}$$

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-5x+\frac{25}{4}=-6+\frac{25}{4}$$

Add -6 to \frac{25}{4}.

$$x^{2}-5x+\frac{25}{4}=\frac{1}{4}$$

Factor x^{2}-5x+\frac{25}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{1}{4}}$$

Simplify.

$$x-\frac{5}{2}=\frac{1}{2} x-\frac{5}{2}=-\frac{1}{2}$$

Add \frac{5}{2} to both sides of the equation.

$$x=3 x=2$$