Solve 450/x=450/x-1-15

$$\frac{ 450 }{ x } = \frac{ 450 }{ x-1 } -15$$

Solve for x

$$x=-5$$

$$x=6$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values 0,1 since division by zero is not defined. Multiply both sides of the equation by x\left(x-1\right), the least common multiple of x,x-1.

$$\left(x-1\right)\times 450=x\times 450+x\left(x-1\right)\left(-15\right)$$

Use the distributive property to multiply x-1 by 450.

$$450x-450=x\times 450+x\left(x-1\right)\left(-15\right)$$

Use the distributive property to multiply x by x-1.

$$450x-450=x\times 450+\left(x^{2}-x\right)\left(-15\right)$$

Use the distributive property to multiply x^{2}-x by -15.

$$450x-450=x\times 450-15x^{2}+15x$$

Combine x\times 450 and 15x to get 465x.

$$450x-450=465x-15x^{2}$$

Subtract 465x from both sides.

$$450x-450-465x=-15x^{2}$$

Combine 450x and -465x to get -15x.

$$-15x-450=-15x^{2}$$

Add 15x^{2} to both sides.

$$-15x-450+15x^{2}=0$$

Divide both sides by 15.

$$-x-30+x^{2}=0$$

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

$$x^{2}-x-30=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-30. To find a and b, set up a system to be solved.

$$a+b=-1 ab=1\left(-30\right)=-30$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -30.

$$1,-30 2,-15 3,-10 5,-6$$

Calculate the sum for each pair.

$$1-30=-29 2-15=-13 3-10=-7 5-6=-1$$

The solution is the pair that gives sum -1.

$$a=-6 b=5$$

Rewrite x^{2}-x-30 as \left(x^{2}-6x\right)+\left(5x-30\right).

$$\left(x^{2}-6x\right)+\left(5x-30\right)$$

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

$$x\left(x-6\right)+5\left(x-6\right)$$

Factor out common term x-6 by using distributive property.

$$\left(x-6\right)\left(x+5\right)$$

To find equation solutions, solve x-6=0 and x+5=0.

$$x=6 x=-5$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,1 since division by zero is not defined. Multiply both sides of the equation by x\left(x-1\right), the least common multiple of x,x-1.

$$\left(x-1\right)\times 450=x\times 450+x\left(x-1\right)\left(-15\right)$$

Use the distributive property to multiply x-1 by 450.

$$450x-450=x\times 450+x\left(x-1\right)\left(-15\right)$$

Use the distributive property to multiply x by x-1.

$$450x-450=x\times 450+\left(x^{2}-x\right)\left(-15\right)$$

Use the distributive property to multiply x^{2}-x by -15.

$$450x-450=x\times 450-15x^{2}+15x$$

Combine x\times 450 and 15x to get 465x.

$$450x-450=465x-15x^{2}$$

Subtract 465x from both sides.

$$450x-450-465x=-15x^{2}$$

Combine 450x and -465x to get -15x.

$$-15x-450=-15x^{2}$$

Add 15x^{2} to both sides.

$$-15x-450+15x^{2}=0$$

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.

$$15x^{2}-15x-450=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 15 for a, -15 for b, and -450 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 15\left(-450\right)}}{2\times 15}$$

Square -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\times 15\left(-450\right)}}{2\times 15}$$

Multiply -4 times 15.

$$x=\frac{-\left(-15\right)±\sqrt{225-60\left(-450\right)}}{2\times 15}$$

Multiply -60 times -450.

$$x=\frac{-\left(-15\right)±\sqrt{225+27000}}{2\times 15}$$

Add 225 to 27000.

$$x=\frac{-\left(-15\right)±\sqrt{27225}}{2\times 15}$$

Take the square root of 27225.

$$x=\frac{-\left(-15\right)±165}{2\times 15}$$

The opposite of -15 is 15.

$$x=\frac{15±165}{2\times 15}$$

Multiply 2 times 15.

$$x=\frac{15±165}{30}$$

Now solve the equation x=\frac{15±165}{30} when ± is plus. Add 15 to 165.

$$x=\frac{180}{30}$$

Divide 180 by 30.

$$x=6$$

Now solve the equation x=\frac{15±165}{30} when ± is minus. Subtract 165 from 15.

$$x=-\frac{150}{30}$$

Divide -150 by 30.

$$x=-5$$

The equation is now solved.

$$x=6 x=-5$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,1 since division by zero is not defined. Multiply both sides of the equation by x\left(x-1\right), the least common multiple of x,x-1.

$$\left(x-1\right)\times 450=x\times 450+x\left(x-1\right)\left(-15\right)$$

Use the distributive property to multiply x-1 by 450.

$$450x-450=x\times 450+x\left(x-1\right)\left(-15\right)$$

Use the distributive property to multiply x by x-1.

$$450x-450=x\times 450+\left(x^{2}-x\right)\left(-15\right)$$

Use the distributive property to multiply x^{2}-x by -15.

$$450x-450=x\times 450-15x^{2}+15x$$

Combine x\times 450 and 15x to get 465x.

$$450x-450=465x-15x^{2}$$

Subtract 465x from both sides.

$$450x-450-465x=-15x^{2}$$

Combine 450x and -465x to get -15x.

$$-15x-450=-15x^{2}$$

Add 15x^{2} to both sides.

$$-15x-450+15x^{2}=0$$

Add 450 to both sides. Anything plus zero gives itself.

$$-15x+15x^{2}=450$$

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.

$$15x^{2}-15x=450$$

Divide both sides by 15.

$$\frac{15x^{2}-15x}{15}=\frac{450}{15}$$

Dividing by 15 undoes the multiplication by 15.

$$x^{2}+\left(-\frac{15}{15}\right)x=\frac{450}{15}$$

Divide -15 by 15.

$$x^{2}-x=\frac{450}{15}$$

Divide 450 by 15.

$$x^{2}-x=30$$

Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=30+\left(-\frac{1}{2}\right)^{2}$$

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-x+\frac{1}{4}=30+\frac{1}{4}$$

Add 30 to \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{121}{4}$$

Factor x^{2}-x+\frac{1}{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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{121}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Simplify.

$$x-\frac{1}{2}=\frac{11}{2} x-\frac{1}{2}=-\frac{11}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$x=6 x=-5$$