Solve 560/x-2-450/x=10

$$\frac { 560 } { x - 2 } - \frac { 450 } { x } = 10$$

Solve for x

$$x=-5$$

$$x=18$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values 0,2 since division by zero is not defined. Multiply both sides of the equation by x\left(x-2\right), the least common multiple of x-2,x.

$$x\times 560-\left(x-2\right)\times 450=10x\left(x-2\right)$$

Use the distributive property to multiply x-2 by 450.

$$x\times 560-\left(450x-900\right)=10x\left(x-2\right)$$

To find the opposite of 450x-900, find the opposite of each term.

$$x\times 560-450x+900=10x\left(x-2\right)$$

Combine x\times 560 and -450x to get 110x.

$$110x+900=10x\left(x-2\right)$$

Use the distributive property to multiply 10x by x-2.

$$110x+900=10x^{2}-20x$$

Subtract 10x^{2} from both sides.

$$110x+900-10x^{2}=-20x$$

Add 20x to both sides.

$$110x+900-10x^{2}+20x=0$$

Combine 110x and 20x to get 130x.

$$130x+900-10x^{2}=0$$

Divide both sides by 10.

$$13x+90-x^{2}=0$$

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

$$-x^{2}+13x+90=0$$

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.

$$a+b=13 ab=-90=-90$$

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 -90.

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

Calculate the sum for each pair.

$$-1+90=89 -2+45=43 -3+30=27 -5+18=13 -6+15=9 -9+10=1$$

The solution is the pair that gives sum 13.

$$a=18 b=-5$$

Rewrite -x^{2}+13x+90 as \left(-x^{2}+18x\right)+\left(-5x+90\right).

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

Factor out -x in the first and -5 in the second group.

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

Factor out common term x-18 by using distributive property.

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

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

$$x=18 x=-5$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,2 since division by zero is not defined. Multiply both sides of the equation by x\left(x-2\right), the least common multiple of x-2,x.

$$x\times 560-\left(x-2\right)\times 450=10x\left(x-2\right)$$

Use the distributive property to multiply x-2 by 450.

$$x\times 560-\left(450x-900\right)=10x\left(x-2\right)$$

To find the opposite of 450x-900, find the opposite of each term.

$$x\times 560-450x+900=10x\left(x-2\right)$$

Combine x\times 560 and -450x to get 110x.

$$110x+900=10x\left(x-2\right)$$

Use the distributive property to multiply 10x by x-2.

$$110x+900=10x^{2}-20x$$

Subtract 10x^{2} from both sides.

$$110x+900-10x^{2}=-20x$$

Add 20x to both sides.

$$110x+900-10x^{2}+20x=0$$

Combine 110x and 20x to get 130x.

$$130x+900-10x^{2}=0$$

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.

$$-10x^{2}+130x+900=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 130 for b, and 900 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-130±\sqrt{130^{2}-4\left(-10\right)\times 900}}{2\left(-10\right)}$$

Square 130.

$$x=\frac{-130±\sqrt{16900-4\left(-10\right)\times 900}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-130±\sqrt{16900+40\times 900}}{2\left(-10\right)}$$

Multiply 40 times 900.

$$x=\frac{-130±\sqrt{16900+36000}}{2\left(-10\right)}$$

Add 16900 to 36000.

$$x=\frac{-130±\sqrt{52900}}{2\left(-10\right)}$$

Take the square root of 52900.

$$x=\frac{-130±230}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-130±230}{-20}$$

Now solve the equation x=\frac{-130±230}{-20} when ± is plus. Add -130 to 230.

$$x=\frac{100}{-20}$$

Divide 100 by -20.

$$x=-5$$

Now solve the equation x=\frac{-130±230}{-20} when ± is minus. Subtract 230 from -130.

$$x=-\frac{360}{-20}$$

Divide -360 by -20.

$$x=18$$

The equation is now solved.

$$x=-5 x=18$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,2 since division by zero is not defined. Multiply both sides of the equation by x\left(x-2\right), the least common multiple of x-2,x.

$$x\times 560-\left(x-2\right)\times 450=10x\left(x-2\right)$$

Use the distributive property to multiply x-2 by 450.

$$x\times 560-\left(450x-900\right)=10x\left(x-2\right)$$

To find the opposite of 450x-900, find the opposite of each term.

$$x\times 560-450x+900=10x\left(x-2\right)$$

Combine x\times 560 and -450x to get 110x.

$$110x+900=10x\left(x-2\right)$$

Use the distributive property to multiply 10x by x-2.

$$110x+900=10x^{2}-20x$$

Subtract 10x^{2} from both sides.

$$110x+900-10x^{2}=-20x$$

Add 20x to both sides.

$$110x+900-10x^{2}+20x=0$$

Combine 110x and 20x to get 130x.

$$130x+900-10x^{2}=0$$

Subtract 900 from both sides. Anything subtracted from zero gives its negation.

$$130x-10x^{2}=-900$$

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.

$$-10x^{2}+130x=-900$$

Divide both sides by -10.

$$\frac{-10x^{2}+130x}{-10}=-\frac{900}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\frac{130}{-10}x=-\frac{900}{-10}$$

Divide 130 by -10.

$$x^{2}-13x=-\frac{900}{-10}$$

Divide -900 by -10.

$$x^{2}-13x=90$$

Divide -13, the coefficient of the x term, by 2 to get -\frac{13}{2}. Then add the square of -\frac{13}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-13x+\left(-\frac{13}{2}\right)^{2}=90+\left(-\frac{13}{2}\right)^{2}$$

Square -\frac{13}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-13x+\frac{169}{4}=90+\frac{169}{4}$$

Add 90 to \frac{169}{4}.

$$x^{2}-13x+\frac{169}{4}=\frac{529}{4}$$

Factor x^{2}-13x+\frac{169}{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}.

$$\left(x-\frac{13}{2}\right)^{2}=\frac{529}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{2}\right)^{2}}=\sqrt{\frac{529}{4}}$$

Simplify.

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

Add \frac{13}{2} to both sides of the equation.

$$x=18 x=-5$$