Solve 7.5/x=7.5/x+15+1/4

$$\frac { 7.5 } { x } = \frac { 7.5 } { x + 15 } + \frac { 1 } { 4 }$$

Solve for x

$$x=-30$$

$$x=15$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -15,0 since division by zero is not defined. Multiply both sides of the equation by 4x\left(x+15\right), the least common multiple of x,x+15,4.

$$\left(4x+60\right)\times 7.5=4x\times 7.5+4x\left(x+15\right)\times \frac{1}{4}$$

Use the distributive property to multiply 4x+60 by 7.5.

$$30x+450=4x\times 7.5+4x\left(x+15\right)\times \frac{1}{4}$$

Multiply 4 and 7.5 to get 30.

$$30x+450=30x+4x\left(x+15\right)\times \frac{1}{4}$$

Multiply 4 and \frac{1}{4} to get 1.

$$30x+450=30x+x\left(x+15\right)$$

Use the distributive property to multiply x by x+15.

$$30x+450=30x+x^{2}+15x$$

Combine 30x and 15x to get 45x.

$$30x+450=45x+x^{2}$$

Subtract 45x from both sides.

$$30x+450-45x=x^{2}$$

Combine 30x and -45x to get -15x.

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

Subtract x^{2} from both sides.

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

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

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+450. To find a and b, set up a system to be solved.

$$a+b=-15 ab=-450=-450$$

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 -450.

$$1,-450 2,-225 3,-150 5,-90 6,-75 9,-50 10,-45 15,-30 18,-25$$

Calculate the sum for each pair.

$$1-450=-449 2-225=-223 3-150=-147 5-90=-85 6-75=-69 9-50=-41 10-45=-35 15-30=-15 18-25=-7$$

The solution is the pair that gives sum -15.

$$a=15 b=-30$$

Rewrite -x^{2}-15x+450 as \left(-x^{2}+15x\right)+\left(-30x+450\right).

$$\left(-x^{2}+15x\right)+\left(-30x+450\right)$$

Factor out x in the first and 30 in the second group.

$$x\left(-x+15\right)+30\left(-x+15\right)$$

Factor out common term -x+15 by using distributive property.

$$\left(-x+15\right)\left(x+30\right)$$

To find equation solutions, solve -x+15=0 and x+30=0.

$$x=15 x=-30$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -15,0 since division by zero is not defined. Multiply both sides of the equation by 4x\left(x+15\right), the least common multiple of x,x+15,4.

$$\left(4x+60\right)\times 7.5=4x\times 7.5+4x\left(x+15\right)\times \frac{1}{4}$$

Use the distributive property to multiply 4x+60 by 7.5.

$$30x+450=4x\times 7.5+4x\left(x+15\right)\times \frac{1}{4}$$

Multiply 4 and 7.5 to get 30.

$$30x+450=30x+4x\left(x+15\right)\times \frac{1}{4}$$

Multiply 4 and \frac{1}{4} to get 1.

$$30x+450=30x+x\left(x+15\right)$$

Use the distributive property to multiply x by x+15.

$$30x+450=30x+x^{2}+15x$$

Combine 30x and 15x to get 45x.

$$30x+450=45x+x^{2}$$

Subtract 45x from both sides.

$$30x+450-45x=x^{2}$$

Combine 30x and -45x to get -15x.

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

Subtract x^{2} from both sides.

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

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.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -15 for b, and 450 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\left(-1\right)\times 450}}{2\left(-1\right)}$$

Square -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\left(-1\right)\times 450}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-15\right)±\sqrt{225+4\times 450}}{2\left(-1\right)}$$

Multiply 4 times 450.

$$x=\frac{-\left(-15\right)±\sqrt{225+1800}}{2\left(-1\right)}$$

Add 225 to 1800.

$$x=\frac{-\left(-15\right)±\sqrt{2025}}{2\left(-1\right)}$$

Take the square root of 2025.

$$x=\frac{-\left(-15\right)±45}{2\left(-1\right)}$$

The opposite of -15 is 15.

$$x=\frac{15±45}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{15±45}{-2}$$

Now solve the equation x=\frac{15±45}{-2} when ± is plus. Add 15 to 45.

$$x=\frac{60}{-2}$$

Divide 60 by -2.

$$x=-30$$

Now solve the equation x=\frac{15±45}{-2} when ± is minus. Subtract 45 from 15.

$$x=-\frac{30}{-2}$$

Divide -30 by -2.

$$x=15$$

The equation is now solved.

$$x=-30 x=15$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -15,0 since division by zero is not defined. Multiply both sides of the equation by 4x\left(x+15\right), the least common multiple of x,x+15,4.

$$\left(4x+60\right)\times 7.5=4x\times 7.5+4x\left(x+15\right)\times \frac{1}{4}$$

Use the distributive property to multiply 4x+60 by 7.5.

$$30x+450=4x\times 7.5+4x\left(x+15\right)\times \frac{1}{4}$$

Multiply 4 and 7.5 to get 30.

$$30x+450=30x+4x\left(x+15\right)\times \frac{1}{4}$$

Multiply 4 and \frac{1}{4} to get 1.

$$30x+450=30x+x\left(x+15\right)$$

Use the distributive property to multiply x by x+15.

$$30x+450=30x+x^{2}+15x$$

Combine 30x and 15x to get 45x.

$$30x+450=45x+x^{2}$$

Subtract 45x from both sides.

$$30x+450-45x=x^{2}$$

Combine 30x and -45x to get -15x.

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

Subtract x^{2} from both sides.

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

Subtract 450 from both sides. Anything subtracted from zero gives its negation.

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

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.

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

Divide both sides by -1.

$$\frac{-x^{2}-15x}{-1}=-\frac{450}{-1}$$

Dividing by -1 undoes the multiplication by -1.

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

Divide -15 by -1.

$$x^{2}+15x=-\frac{450}{-1}$$

Divide -450 by -1.

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

Divide 15, the coefficient of the x term, by 2 to get \frac{15}{2}. Then add the square of \frac{15}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square \frac{15}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+15x+\frac{225}{4}=450+\frac{225}{4}$$

Add 450 to \frac{225}{4}.

$$x^{2}+15x+\frac{225}{4}=\frac{2025}{4}$$

Factor x^{2}+15x+\frac{225}{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}.

$$\left(x+\frac{15}{2}\right)^{2}=\frac{2025}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{15}{2}\right)^{2}}=\sqrt{\frac{2025}{4}}$$

Simplify.

$$x+\frac{15}{2}=\frac{45}{2} x+\frac{15}{2}=-\frac{45}{2}$$

Subtract \frac{15}{2} from both sides of the equation.

$$x=15 x=-30$$