Solve 24/x-3=32x^2/x^2-9-16x/x+3

$$\frac { 24 } { x - 3 } = \frac { 32 x ^ { 2 } } { x ^ { 2 } - 9 } - \frac { 16 x } { x + 3 }$$

Solve for x

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

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -3,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x+3\right), the least common multiple of x-3,x^{2}-9,x+3.

$$\left(x+3\right)\times 24=32x^{2}-\left(x-3\right)\times 16x$$

Use the distributive property to multiply x+3 by 24.

$$24x+72=32x^{2}-\left(x-3\right)\times 16x$$

Use the distributive property to multiply x-3 by 16.

$$24x+72=32x^{2}-\left(16x-48\right)x$$

Use the distributive property to multiply 16x-48 by x.

$$24x+72=32x^{2}-\left(16x^{2}-48x\right)$$

To find the opposite of 16x^{2}-48x, find the opposite of each term.

$$24x+72=32x^{2}-16x^{2}+48x$$

Combine 32x^{2} and -16x^{2} to get 16x^{2}.

$$24x+72=16x^{2}+48x$$

Subtract 16x^{2} from both sides.

$$24x+72-16x^{2}=48x$$

Subtract 48x from both sides.

$$24x+72-16x^{2}-48x=0$$

Combine 24x and -48x to get -24x.

$$-24x+72-16x^{2}=0$$

Divide both sides by 8.

$$-3x+9-2x^{2}=0$$

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

$$-2x^{2}-3x+9=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -2x^{2}+ax+bx+9. To find a and b, set up a system to be solved.

$$a+b=-3 ab=-2\times 9=-18$$

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 -18.

$$1,-18 2,-9 3,-6$$

Calculate the sum for each pair.

$$1-18=-17 2-9=-7 3-6=-3$$

The solution is the pair that gives sum -3.

$$a=3 b=-6$$

Rewrite -2x^{2}-3x+9 as \left(-2x^{2}+3x\right)+\left(-6x+9\right).

$$\left(-2x^{2}+3x\right)+\left(-6x+9\right)$$

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

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

Factor out common term 2x-3 by using distributive property.

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

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

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

Variable x cannot be equal to -3.

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

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -3,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x+3\right), the least common multiple of x-3,x^{2}-9,x+3.

$$\left(x+3\right)\times 24=32x^{2}-\left(x-3\right)\times 16x$$

Use the distributive property to multiply x+3 by 24.

$$24x+72=32x^{2}-\left(x-3\right)\times 16x$$

Use the distributive property to multiply x-3 by 16.

$$24x+72=32x^{2}-\left(16x-48\right)x$$

Use the distributive property to multiply 16x-48 by x.

$$24x+72=32x^{2}-\left(16x^{2}-48x\right)$$

To find the opposite of 16x^{2}-48x, find the opposite of each term.

$$24x+72=32x^{2}-16x^{2}+48x$$

Combine 32x^{2} and -16x^{2} to get 16x^{2}.

$$24x+72=16x^{2}+48x$$

Subtract 16x^{2} from both sides.

$$24x+72-16x^{2}=48x$$

Subtract 48x from both sides.

$$24x+72-16x^{2}-48x=0$$

Combine 24x and -48x to get -24x.

$$-24x+72-16x^{2}=0$$

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.

$$-16x^{2}-24x+72=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, -24 for b, and 72 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\left(-16\right)\times 72}}{2\left(-16\right)}$$

Square -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\left(-16\right)\times 72}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$x=\frac{-\left(-24\right)±\sqrt{576+64\times 72}}{2\left(-16\right)}$$

Multiply 64 times 72.

$$x=\frac{-\left(-24\right)±\sqrt{576+4608}}{2\left(-16\right)}$$

Add 576 to 4608.

$$x=\frac{-\left(-24\right)±\sqrt{5184}}{2\left(-16\right)}$$

Take the square root of 5184.

$$x=\frac{-\left(-24\right)±72}{2\left(-16\right)}$$

The opposite of -24 is 24.

$$x=\frac{24±72}{2\left(-16\right)}$$

Multiply 2 times -16.

$$x=\frac{24±72}{-32}$$

Now solve the equation x=\frac{24±72}{-32} when ± is plus. Add 24 to 72.

$$x=\frac{96}{-32}$$

Divide 96 by -32.

$$x=-3$$

Now solve the equation x=\frac{24±72}{-32} when ± is minus. Subtract 72 from 24.

$$x=-\frac{48}{-32}$$

Reduce the fraction \frac{-48}{-32} to lowest terms by extracting and canceling out 16.

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

The equation is now solved.

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

Variable x cannot be equal to -3.

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

Steps for Completing the Square

Variable x cannot be equal to any of the values -3,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x+3\right), the least common multiple of x-3,x^{2}-9,x+3.

$$\left(x+3\right)\times 24=32x^{2}-\left(x-3\right)\times 16x$$

Use the distributive property to multiply x+3 by 24.

$$24x+72=32x^{2}-\left(x-3\right)\times 16x$$

Use the distributive property to multiply x-3 by 16.

$$24x+72=32x^{2}-\left(16x-48\right)x$$

Use the distributive property to multiply 16x-48 by x.

$$24x+72=32x^{2}-\left(16x^{2}-48x\right)$$

To find the opposite of 16x^{2}-48x, find the opposite of each term.

$$24x+72=32x^{2}-16x^{2}+48x$$

Combine 32x^{2} and -16x^{2} to get 16x^{2}.

$$24x+72=16x^{2}+48x$$

Subtract 16x^{2} from both sides.

$$24x+72-16x^{2}=48x$$

Subtract 48x from both sides.

$$24x+72-16x^{2}-48x=0$$

Combine 24x and -48x to get -24x.

$$-24x+72-16x^{2}=0$$

Subtract 72 from both sides. Anything subtracted from zero gives its negation.

$$-24x-16x^{2}=-72$$

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.

$$-16x^{2}-24x=-72$$

Divide both sides by -16.

$$\frac{-16x^{2}-24x}{-16}=-\frac{72}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$x^{2}+\left(-\frac{24}{-16}\right)x=-\frac{72}{-16}$$

Reduce the fraction \frac{-24}{-16} to lowest terms by extracting and canceling out 8.

$$x^{2}+\frac{3}{2}x=-\frac{72}{-16}$$

Reduce the fraction \frac{-72}{-16} to lowest terms by extracting and canceling out 8.

$$x^{2}+\frac{3}{2}x=\frac{9}{2}$$

Divide \frac{3}{2}, the coefficient of the x term, by 2 to get \frac{3}{4}. Then add the square of \frac{3}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{3}{2}x+\left(\frac{3}{4}\right)^{2}=\frac{9}{2}+\left(\frac{3}{4}\right)^{2}$$

Square \frac{3}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{3}{2}x+\frac{9}{16}=\frac{9}{2}+\frac{9}{16}$$

Add \frac{9}{2} to \frac{9}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{3}{2}x+\frac{9}{16}=\frac{81}{16}$$

Factor x^{2}+\frac{3}{2}x+\frac{9}{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}.

$$\left(x+\frac{3}{4}\right)^{2}=\frac{81}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{4}\right)^{2}}=\sqrt{\frac{81}{16}}$$

Simplify.

$$x+\frac{3}{4}=\frac{9}{4} x+\frac{3}{4}=-\frac{9}{4}$$

Subtract \frac{3}{4} from both sides of the equation.

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

Variable x cannot be equal to -3.

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