Solve 18/x-46/x-5=1

$$\frac { 18 } { x } - \frac { 46 } { x - 5 } = 1$$

Solve for x

$$x=-18$$

$$x=-5$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values 0,5 since division by zero is not defined. Multiply both sides of the equation by x\left(x-5\right), the least common multiple of x,x-5.

$$\left(x-5\right)\times 18-x\times 46=x\left(x-5\right)$$

Use the distributive property to multiply x-5 by 18.

$$18x-90-x\times 46=x\left(x-5\right)$$

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

$$18x-90-x\times 46=x^{2}-5x$$

Subtract x^{2} from both sides.

$$18x-90-x\times 46-x^{2}=-5x$$

Add 5x to both sides.

$$18x-90-x\times 46-x^{2}+5x=0$$

Combine 18x and 5x to get 23x.

$$23x-90-x\times 46-x^{2}=0$$

Multiply -1 and 46 to get -46.

$$23x-90-46x-x^{2}=0$$

Combine 23x and -46x to get -23x.

$$-23x-90-x^{2}=0$$

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

$$-x^{2}-23x-90=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-90. To find a and b, set up a system to be solved.

$$a+b=-23 ab=-\left(-90\right)=90$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 90.

$$-1,-90 -2,-45 -3,-30 -5,-18 -6,-15 -9,-10$$

Calculate the sum for each pair.

$$-1-90=-91 -2-45=-47 -3-30=-33 -5-18=-23 -6-15=-21 -9-10=-19$$

The solution is the pair that gives sum -23.

$$a=-5 b=-18$$

Rewrite -x^{2}-23x-90 as \left(-x^{2}-5x\right)+\left(-18x-90\right).

$$\left(-x^{2}-5x\right)+\left(-18x-90\right)$$

Factor out x in the first and 18 in the second group.

$$x\left(-x-5\right)+18\left(-x-5\right)$$

Factor out common term -x-5 by using distributive property.

$$\left(-x-5\right)\left(x+18\right)$$

To find equation solutions, solve -x-5=0 and x+18=0.

$$x=-5 x=-18$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,5 since division by zero is not defined. Multiply both sides of the equation by x\left(x-5\right), the least common multiple of x,x-5.

$$\left(x-5\right)\times 18-x\times 46=x\left(x-5\right)$$

Use the distributive property to multiply x-5 by 18.

$$18x-90-x\times 46=x\left(x-5\right)$$

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

$$18x-90-x\times 46=x^{2}-5x$$

Subtract x^{2} from both sides.

$$18x-90-x\times 46-x^{2}=-5x$$

Add 5x to both sides.

$$18x-90-x\times 46-x^{2}+5x=0$$

Combine 18x and 5x to get 23x.

$$23x-90-x\times 46-x^{2}=0$$

Multiply -1 and 46 to get -46.

$$23x-90-46x-x^{2}=0$$

Combine 23x and -46x to get -23x.

$$-23x-90-x^{2}=0$$

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.

$$-x^{2}-23x-90=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -23 for b, and -90 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-23\right)±\sqrt{\left(-23\right)^{2}-4\left(-1\right)\left(-90\right)}}{2\left(-1\right)}$$

Square -23.

$$x=\frac{-\left(-23\right)±\sqrt{529-4\left(-1\right)\left(-90\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-23\right)±\sqrt{529+4\left(-90\right)}}{2\left(-1\right)}$$

Multiply 4 times -90.

$$x=\frac{-\left(-23\right)±\sqrt{529-360}}{2\left(-1\right)}$$

Add 529 to -360.

$$x=\frac{-\left(-23\right)±\sqrt{169}}{2\left(-1\right)}$$

Take the square root of 169.

$$x=\frac{-\left(-23\right)±13}{2\left(-1\right)}$$

The opposite of -23 is 23.

$$x=\frac{23±13}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{23±13}{-2}$$

Now solve the equation x=\frac{23±13}{-2} when ± is plus. Add 23 to 13.

$$x=\frac{36}{-2}$$

Divide 36 by -2.

$$x=-18$$

Now solve the equation x=\frac{23±13}{-2} when ± is minus. Subtract 13 from 23.

$$x=\frac{10}{-2}$$

Divide 10 by -2.

$$x=-5$$

The equation is now solved.

$$x=-18 x=-5$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,5 since division by zero is not defined. Multiply both sides of the equation by x\left(x-5\right), the least common multiple of x,x-5.

$$\left(x-5\right)\times 18-x\times 46=x\left(x-5\right)$$

Use the distributive property to multiply x-5 by 18.

$$18x-90-x\times 46=x\left(x-5\right)$$

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

$$18x-90-x\times 46=x^{2}-5x$$

Subtract x^{2} from both sides.

$$18x-90-x\times 46-x^{2}=-5x$$

Add 5x to both sides.

$$18x-90-x\times 46-x^{2}+5x=0$$

Combine 18x and 5x to get 23x.

$$23x-90-x\times 46-x^{2}=0$$

Add 90 to both sides. Anything plus zero gives itself.

$$23x-x\times 46-x^{2}=90$$

Multiply -1 and 46 to get -46.

$$23x-46x-x^{2}=90$$

Combine 23x and -46x to get -23x.

$$-23x-x^{2}=90$$

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.

$$-x^{2}-23x=90$$

Divide both sides by -1.

$$\frac{-x^{2}-23x}{-1}=\frac{90}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{23}{-1}\right)x=\frac{90}{-1}$$

Divide -23 by -1.

$$x^{2}+23x=\frac{90}{-1}$$

Divide 90 by -1.

$$x^{2}+23x=-90$$

Divide 23, the coefficient of the x term, by 2 to get \frac{23}{2}. Then add the square of \frac{23}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+23x+\left(\frac{23}{2}\right)^{2}=-90+\left(\frac{23}{2}\right)^{2}$$

Square \frac{23}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+23x+\frac{529}{4}=-90+\frac{529}{4}$$

Add -90 to \frac{529}{4}.

$$x^{2}+23x+\frac{529}{4}=\frac{169}{4}$$

Factor x^{2}+23x+\frac{529}{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}.

$$\left(x+\frac{23}{2}\right)^{2}=\frac{169}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{23}{2}\right)^{2}}=\sqrt{\frac{169}{4}}$$

Simplify.

$$x+\frac{23}{2}=\frac{13}{2} x+\frac{23}{2}=-\frac{13}{2}$$

Subtract \frac{23}{2} from both sides of the equation.

$$x=-5 x=-18$$