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

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

Solve for x

$$x=-5$$

$$x=6$$

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-1,x.

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

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

$$x\times 450+\left(x^{2}-x\right)\left(-15\right)=\left(x-1\right)\times 450$$

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

$$x\times 450-15x^{2}+15x=\left(x-1\right)\times 450$$

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

$$465x-15x^{2}=\left(x-1\right)\times 450$$

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

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

Subtract 450x from both sides.

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

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

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

Add 450 to both sides.

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

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}.

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

Square 15.

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

Multiply -4 times -15.

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

Multiply 60 times 450.

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

Add 225 to 27000.

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

Take the square root of 27225.

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

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.

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

Divide 150 by -30.

$$x=-5$$

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

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

Divide -180 by -30.

$$x=6$$

The equation is now solved.

$$x=-5 x=6$$

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-1,x.

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

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

$$x\times 450+\left(x^{2}-x\right)\left(-15\right)=\left(x-1\right)\times 450$$

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

$$x\times 450-15x^{2}+15x=\left(x-1\right)\times 450$$

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

$$465x-15x^{2}=\left(x-1\right)\times 450$$

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

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

Subtract 450x from both sides.

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

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

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

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.

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

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.

$$x=6 x=-5$$