Solve 10/x-3-8/x=1

$$\frac { 10 } { x - 3 } - \frac { 8 } { x } = 1$$

Solve for x

$$x=-3$$

$$x=8$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values 0,3 since division by zero is not defined. Multiply both sides of the equation by x\left(x-3\right), the least common multiple of x-3,x.

$$x\times 10-\left(x-3\right)\times 8=x\left(x-3\right)$$

Use the distributive property to multiply x-3 by 8.

$$x\times 10-\left(8x-24\right)=x\left(x-3\right)$$

To find the opposite of 8x-24, find the opposite of each term.

$$x\times 10-8x+24=x\left(x-3\right)$$

Combine x\times 10 and -8x to get 2x.

$$2x+24=x\left(x-3\right)$$

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

$$2x+24=x^{2}-3x$$

Subtract x^{2} from both sides.

$$2x+24-x^{2}=-3x$$

Add 3x to both sides.

$$2x+24-x^{2}+3x=0$$

Combine 2x and 3x to get 5x.

$$5x+24-x^{2}=0$$

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

$$-x^{2}+5x+24=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+24. To find a and b, set up a system to be solved.

$$a+b=5 ab=-24=-24$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -24.

$$-1,24 -2,12 -3,8 -4,6$$

Calculate the sum for each pair.

$$-1+24=23 -2+12=10 -3+8=5 -4+6=2$$

The solution is the pair that gives sum 5.

$$a=8 b=-3$$

Rewrite -x^{2}+5x+24 as \left(-x^{2}+8x\right)+\left(-3x+24\right).

$$\left(-x^{2}+8x\right)+\left(-3x+24\right)$$

Factor out -x in the first and -3 in the second group.

$$-x\left(x-8\right)-3\left(x-8\right)$$

Factor out common term x-8 by using distributive property.

$$\left(x-8\right)\left(-x-3\right)$$

To find equation solutions, solve x-8=0 and -x-3=0.

$$x=8 x=-3$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,3 since division by zero is not defined. Multiply both sides of the equation by x\left(x-3\right), the least common multiple of x-3,x.

$$x\times 10-\left(x-3\right)\times 8=x\left(x-3\right)$$

Use the distributive property to multiply x-3 by 8.

$$x\times 10-\left(8x-24\right)=x\left(x-3\right)$$

To find the opposite of 8x-24, find the opposite of each term.

$$x\times 10-8x+24=x\left(x-3\right)$$

Combine x\times 10 and -8x to get 2x.

$$2x+24=x\left(x-3\right)$$

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

$$2x+24=x^{2}-3x$$

Subtract x^{2} from both sides.

$$2x+24-x^{2}=-3x$$

Add 3x to both sides.

$$2x+24-x^{2}+3x=0$$

Combine 2x and 3x to get 5x.

$$5x+24-x^{2}=0$$

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.

$$-x^{2}+5x+24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 5 for b, and 24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\left(-1\right)\times 24}}{2\left(-1\right)}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\left(-1\right)\times 24}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-5±\sqrt{25+4\times 24}}{2\left(-1\right)}$$

Multiply 4 times 24.

$$x=\frac{-5±\sqrt{25+96}}{2\left(-1\right)}$$

Add 25 to 96.

$$x=\frac{-5±\sqrt{121}}{2\left(-1\right)}$$

Take the square root of 121.

$$x=\frac{-5±11}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-5±11}{-2}$$

Now solve the equation x=\frac{-5±11}{-2} when ± is plus. Add -5 to 11.

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

Divide 6 by -2.

$$x=-3$$

Now solve the equation x=\frac{-5±11}{-2} when ± is minus. Subtract 11 from -5.

$$x=-\frac{16}{-2}$$

Divide -16 by -2.

$$x=8$$

The equation is now solved.

$$x=-3 x=8$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,3 since division by zero is not defined. Multiply both sides of the equation by x\left(x-3\right), the least common multiple of x-3,x.

$$x\times 10-\left(x-3\right)\times 8=x\left(x-3\right)$$

Use the distributive property to multiply x-3 by 8.

$$x\times 10-\left(8x-24\right)=x\left(x-3\right)$$

To find the opposite of 8x-24, find the opposite of each term.

$$x\times 10-8x+24=x\left(x-3\right)$$

Combine x\times 10 and -8x to get 2x.

$$2x+24=x\left(x-3\right)$$

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

$$2x+24=x^{2}-3x$$

Subtract x^{2} from both sides.

$$2x+24-x^{2}=-3x$$

Add 3x to both sides.

$$2x+24-x^{2}+3x=0$$

Combine 2x and 3x to get 5x.

$$5x+24-x^{2}=0$$

Subtract 24 from both sides. Anything subtracted from zero gives its negation.

$$5x-x^{2}=-24$$

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.

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

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

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

Divide 5 by -1.

$$x^{2}-5x=-\frac{24}{-1}$$

Divide -24 by -1.

$$x^{2}-5x=24$$

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.

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=24+\left(-\frac{5}{2}\right)^{2}$$

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

$$x^{2}-5x+\frac{25}{4}=24+\frac{25}{4}$$

Add 24 to \frac{25}{4}.

$$x^{2}-5x+\frac{25}{4}=\frac{121}{4}$$

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}.

$$\left(x-\frac{5}{2}\right)^{2}=\frac{121}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Simplify.

$$x-\frac{5}{2}=\frac{11}{2} x-\frac{5}{2}=-\frac{11}{2}$$

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

$$x=8 x=-3$$