Solve 10/x-12/x-3+4=0

$$\frac { 10 } { x } - \frac { 12 } { x - 3 } + 4 = 0$$

Solve for x

$$x = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

$$x=5$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values 0,3 since division by zero is not defined. Multiply both sides of the equation by x\left(x-3\right), the least common multiple of x,x-3.

$$\left(x-3\right)\times 10-x\times 12+x\left(x-3\right)\times 4=0$$

Use the distributive property to multiply x-3 by 10.

$$10x-30-x\times 12+x\left(x-3\right)\times 4=0$$

Use the distributive property to multiply x by x-3.

$$10x-30-x\times 12+\left(x^{2}-3x\right)\times 4=0$$

Use the distributive property to multiply x^{2}-3x by 4.

$$10x-30-x\times 12+4x^{2}-12x=0$$

Combine 10x and -12x to get -2x.

$$-2x-30-x\times 12+4x^{2}=0$$

Multiply -1 and 12 to get -12.

$$-2x-30-12x+4x^{2}=0$$

Combine -2x and -12x to get -14x.

$$-14x-30+4x^{2}=0$$

Divide both sides by 2.

$$-7x-15+2x^{2}=0$$

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

$$2x^{2}-7x-15=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 2x^{2}+ax+bx-15. To find a and b, set up a system to be solved.

$$a+b=-7 ab=2\left(-15\right)=-30$$

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 -7.

$$a=-10 b=3$$

Rewrite 2x^{2}-7x-15 as \left(2x^{2}-10x\right)+\left(3x-15\right).

$$\left(2x^{2}-10x\right)+\left(3x-15\right)$$

Factor out 2x in the first and 3 in the second group.

$$2x\left(x-5\right)+3\left(x-5\right)$$

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

$$\left(x-5\right)\left(2x+3\right)$$

To find equation solutions, solve x-5=0 and 2x+3=0.

$$x=5 x=-\frac{3}{2}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,3 since division by zero is not defined. Multiply both sides of the equation by x\left(x-3\right), the least common multiple of x,x-3.

$$\left(x-3\right)\times 10-x\times 12+x\left(x-3\right)\times 4=0$$

Use the distributive property to multiply x-3 by 10.

$$10x-30-x\times 12+x\left(x-3\right)\times 4=0$$

Use the distributive property to multiply x by x-3.

$$10x-30-x\times 12+\left(x^{2}-3x\right)\times 4=0$$

Use the distributive property to multiply x^{2}-3x by 4.

$$10x-30-x\times 12+4x^{2}-12x=0$$

Combine 10x and -12x to get -2x.

$$-2x-30-x\times 12+4x^{2}=0$$

Multiply -1 and 12 to get -12.

$$-2x-30-12x+4x^{2}=0$$

Combine -2x and -12x to get -14x.

$$-14x-30+4x^{2}=0$$

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.

$$4x^{2}-14x-30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -14 for b, and -30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 4\left(-30\right)}}{2\times 4}$$

Square -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 4\left(-30\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-14\right)±\sqrt{196-16\left(-30\right)}}{2\times 4}$$

Multiply -16 times -30.

$$x=\frac{-\left(-14\right)±\sqrt{196+480}}{2\times 4}$$

Add 196 to 480.

$$x=\frac{-\left(-14\right)±\sqrt{676}}{2\times 4}$$

Take the square root of 676.

$$x=\frac{-\left(-14\right)±26}{2\times 4}$$

The opposite of -14 is 14.

$$x=\frac{14±26}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{14±26}{8}$$

Now solve the equation x=\frac{14±26}{8} when ± is plus. Add 14 to 26.

$$x=\frac{40}{8}$$

Divide 40 by 8.

$$x=5$$

Now solve the equation x=\frac{14±26}{8} when ± is minus. Subtract 26 from 14.

$$x=-\frac{12}{8}$$

Reduce the fraction \frac{-12}{8} to lowest terms by extracting and canceling out 4.

$$x=-\frac{3}{2}$$

The equation is now solved.

$$x=5 x=-\frac{3}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,3 since division by zero is not defined. Multiply both sides of the equation by x\left(x-3\right), the least common multiple of x,x-3.

$$\left(x-3\right)\times 10-x\times 12+x\left(x-3\right)\times 4=0$$

Use the distributive property to multiply x-3 by 10.

$$10x-30-x\times 12+x\left(x-3\right)\times 4=0$$

Use the distributive property to multiply x by x-3.

$$10x-30-x\times 12+\left(x^{2}-3x\right)\times 4=0$$

Use the distributive property to multiply x^{2}-3x by 4.

$$10x-30-x\times 12+4x^{2}-12x=0$$

Combine 10x and -12x to get -2x.

$$-2x-30-x\times 12+4x^{2}=0$$

Add 30 to both sides. Anything plus zero gives itself.

$$-2x-x\times 12+4x^{2}=30$$

Multiply -1 and 12 to get -12.

$$-2x-12x+4x^{2}=30$$

Combine -2x and -12x to get -14x.

$$-14x+4x^{2}=30$$

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.

$$4x^{2}-14x=30$$

Divide both sides by 4.

$$\frac{4x^{2}-14x}{4}=\frac{30}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\left(-\frac{14}{4}\right)x=\frac{30}{4}$$

Reduce the fraction \frac{-14}{4} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{7}{2}x=\frac{30}{4}$$

Reduce the fraction \frac{30}{4} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{7}{2}x=\frac{15}{2}$$

Divide -\frac{7}{2}, the coefficient of the x term, by 2 to get -\frac{7}{4}. Then add the square of -\frac{7}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{7}{2}x+\left(-\frac{7}{4}\right)^{2}=\frac{15}{2}+\left(-\frac{7}{4}\right)^{2}$$

Square -\frac{7}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=\frac{15}{2}+\frac{49}{16}$$

Add \frac{15}{2} to \frac{49}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=\frac{169}{16}$$

Factor x^{2}-\frac{7}{2}x+\frac{49}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{4}\right)^{2}=\frac{169}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{4}\right)^{2}}=\sqrt{\frac{169}{16}}$$

Simplify.

$$x-\frac{7}{4}=\frac{13}{4} x-\frac{7}{4}=-\frac{13}{4}$$

Add \frac{7}{4} to both sides of the equation.

$$x=5 x=-\frac{3}{2}$$