Solve 32/24-x-32/24+x=1

$$\frac { 32 } { 24 - x } - \frac { 32 } { 24 + x } = 1$$

Solve for x

$$x=-72$$

$$x=8$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -24,24 since division by zero is not defined. Multiply both sides of the equation by \left(x-24\right)\left(x+24\right), the least common multiple of 24-x,24+x.

$$-\left(24+x\right)\times 32-\left(x-24\right)\times 32=\left(x-24\right)\left(x+24\right)$$

To find the opposite of 24+x, find the opposite of each term.

$$\left(-24-x\right)\times 32-\left(x-24\right)\times 32=\left(x-24\right)\left(x+24\right)$$

Use the distributive property to multiply -24-x by 32.

$$-768-32x-\left(x-24\right)\times 32=\left(x-24\right)\left(x+24\right)$$

Use the distributive property to multiply x-24 by 32.

$$-768-32x-\left(32x-768\right)=\left(x-24\right)\left(x+24\right)$$

To find the opposite of 32x-768, find the opposite of each term.

$$-768-32x-32x+768=\left(x-24\right)\left(x+24\right)$$

Combine -32x and -32x to get -64x.

$$-768-64x+768=\left(x-24\right)\left(x+24\right)$$

Add -768 and 768 to get 0.

$$-64x=\left(x-24\right)\left(x+24\right)$$

Consider \left(x-24\right)\left(x+24\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 24.

$$-64x=x^{2}-576$$

Subtract x^{2} from both sides.

$$-64x-x^{2}=-576$$

Add 576 to both sides.

$$-64x-x^{2}+576=0$$

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.

$$-x^{2}-64x+576=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -64 for b, and 576 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-64\right)±\sqrt{\left(-64\right)^{2}-4\left(-1\right)\times 576}}{2\left(-1\right)}$$

Square -64.

$$x=\frac{-\left(-64\right)±\sqrt{4096-4\left(-1\right)\times 576}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-64\right)±\sqrt{4096+4\times 576}}{2\left(-1\right)}$$

Multiply 4 times 576.

$$x=\frac{-\left(-64\right)±\sqrt{4096+2304}}{2\left(-1\right)}$$

Add 4096 to 2304.

$$x=\frac{-\left(-64\right)±\sqrt{6400}}{2\left(-1\right)}$$

Take the square root of 6400.

$$x=\frac{-\left(-64\right)±80}{2\left(-1\right)}$$

The opposite of -64 is 64.

$$x=\frac{64±80}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{64±80}{-2}$$

Now solve the equation x=\frac{64±80}{-2} when ± is plus. Add 64 to 80.

$$x=\frac{144}{-2}$$

Divide 144 by -2.

$$x=-72$$

Now solve the equation x=\frac{64±80}{-2} when ± is minus. Subtract 80 from 64.

$$x=-\frac{16}{-2}$$

Divide -16 by -2.

$$x=8$$

The equation is now solved.

$$x=-72 x=8$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -24,24 since division by zero is not defined. Multiply both sides of the equation by \left(x-24\right)\left(x+24\right), the least common multiple of 24-x,24+x.

$$-\left(24+x\right)\times 32-\left(x-24\right)\times 32=\left(x-24\right)\left(x+24\right)$$

To find the opposite of 24+x, find the opposite of each term.

$$\left(-24-x\right)\times 32-\left(x-24\right)\times 32=\left(x-24\right)\left(x+24\right)$$

Use the distributive property to multiply -24-x by 32.

$$-768-32x-\left(x-24\right)\times 32=\left(x-24\right)\left(x+24\right)$$

Use the distributive property to multiply x-24 by 32.

$$-768-32x-\left(32x-768\right)=\left(x-24\right)\left(x+24\right)$$

To find the opposite of 32x-768, find the opposite of each term.

$$-768-32x-32x+768=\left(x-24\right)\left(x+24\right)$$

Combine -32x and -32x to get -64x.

$$-768-64x+768=\left(x-24\right)\left(x+24\right)$$

Add -768 and 768 to get 0.

$$-64x=\left(x-24\right)\left(x+24\right)$$

Consider \left(x-24\right)\left(x+24\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 24.

$$-64x=x^{2}-576$$

Subtract x^{2} from both sides.

$$-64x-x^{2}=-576$$

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.

$$-x^{2}-64x=-576$$

Divide both sides by -1.

$$\frac{-x^{2}-64x}{-1}=-\frac{576}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{64}{-1}\right)x=-\frac{576}{-1}$$

Divide -64 by -1.

$$x^{2}+64x=-\frac{576}{-1}$$

Divide -576 by -1.

$$x^{2}+64x=576$$

Divide 64, the coefficient of the x term, by 2 to get 32. Then add the square of 32 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+64x+32^{2}=576+32^{2}$$

Square 32.

$$x^{2}+64x+1024=576+1024$$

Add 576 to 1024.

$$x^{2}+64x+1024=1600$$

Factor x^{2}+64x+1024. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+32\right)^{2}=1600$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+32\right)^{2}}=\sqrt{1600}$$

Simplify.

$$x+32=40 x+32=-40$$

Subtract 32 from both sides of the equation.

$$x=8 x=-72$$