Solve 7x/x+4-7=x-22/x-4

$$\frac { 7 x } { x + 4 } - 7 = \frac { x - 22 } { x - 4 }$$

Solve for x

$$x=-20$$

$$x=10$$

Steps Using the Quadratic Formula

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

$$\left(x-4\right)\times 7x+\left(x-4\right)\left(x+4\right)\left(-7\right)=\left(x+4\right)\left(x-22\right)$$

Use the distributive property to multiply x-4 by 7.

$$\left(7x-28\right)x+\left(x-4\right)\left(x+4\right)\left(-7\right)=\left(x+4\right)\left(x-22\right)$$

Use the distributive property to multiply 7x-28 by x.

$$7x^{2}-28x+\left(x-4\right)\left(x+4\right)\left(-7\right)=\left(x+4\right)\left(x-22\right)$$

Use the distributive property to multiply x-4 by x+4 and combine like terms.

$$7x^{2}-28x+\left(x^{2}-16\right)\left(-7\right)=\left(x+4\right)\left(x-22\right)$$

Use the distributive property to multiply x^{2}-16 by -7.

$$7x^{2}-28x-7x^{2}+112=\left(x+4\right)\left(x-22\right)$$

Combine 7x^{2} and -7x^{2} to get 0.

$$-28x+112=\left(x+4\right)\left(x-22\right)$$

Use the distributive property to multiply x+4 by x-22 and combine like terms.

$$-28x+112=x^{2}-18x-88$$

Subtract x^{2} from both sides.

$$-28x+112-x^{2}=-18x-88$$

Add 18x to both sides.

$$-28x+112-x^{2}+18x=-88$$

Combine -28x and 18x to get -10x.

$$-10x+112-x^{2}=-88$$

Add 88 to both sides.

$$-10x+112-x^{2}+88=0$$

Add 112 and 88 to get 200.

$$-10x+200-x^{2}=0$$

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.

$$-x^{2}-10x+200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -10 for b, and 200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\left(-1\right)\times 200}}{2\left(-1\right)}$$

Square -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\left(-1\right)\times 200}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-10\right)±\sqrt{100+4\times 200}}{2\left(-1\right)}$$

Multiply 4 times 200.

$$x=\frac{-\left(-10\right)±\sqrt{100+800}}{2\left(-1\right)}$$

Add 100 to 800.

$$x=\frac{-\left(-10\right)±\sqrt{900}}{2\left(-1\right)}$$

Take the square root of 900.

$$x=\frac{-\left(-10\right)±30}{2\left(-1\right)}$$

The opposite of -10 is 10.

$$x=\frac{10±30}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{10±30}{-2}$$

Now solve the equation x=\frac{10±30}{-2} when ± is plus. Add 10 to 30.

$$x=\frac{40}{-2}$$

Divide 40 by -2.

$$x=-20$$

Now solve the equation x=\frac{10±30}{-2} when ± is minus. Subtract 30 from 10.

$$x=-\frac{20}{-2}$$

Divide -20 by -2.

$$x=10$$

The equation is now solved.

$$x=-20 x=10$$

Steps for Completing the Square

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

$$\left(x-4\right)\times 7x+\left(x-4\right)\left(x+4\right)\left(-7\right)=\left(x+4\right)\left(x-22\right)$$

Use the distributive property to multiply x-4 by 7.

$$\left(7x-28\right)x+\left(x-4\right)\left(x+4\right)\left(-7\right)=\left(x+4\right)\left(x-22\right)$$

Use the distributive property to multiply 7x-28 by x.

$$7x^{2}-28x+\left(x-4\right)\left(x+4\right)\left(-7\right)=\left(x+4\right)\left(x-22\right)$$

Use the distributive property to multiply x-4 by x+4 and combine like terms.

$$7x^{2}-28x+\left(x^{2}-16\right)\left(-7\right)=\left(x+4\right)\left(x-22\right)$$

Use the distributive property to multiply x^{2}-16 by -7.

$$7x^{2}-28x-7x^{2}+112=\left(x+4\right)\left(x-22\right)$$

Combine 7x^{2} and -7x^{2} to get 0.

$$-28x+112=\left(x+4\right)\left(x-22\right)$$

Use the distributive property to multiply x+4 by x-22 and combine like terms.

$$-28x+112=x^{2}-18x-88$$

Subtract x^{2} from both sides.

$$-28x+112-x^{2}=-18x-88$$

Add 18x to both sides.

$$-28x+112-x^{2}+18x=-88$$

Combine -28x and 18x to get -10x.

$$-10x+112-x^{2}=-88$$

Subtract 112 from both sides.

$$-10x-x^{2}=-88-112$$

Subtract 112 from -88 to get -200.

$$-10x-x^{2}=-200$$

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.

$$-x^{2}-10x=-200$$

Divide both sides by -1.

$$\frac{-x^{2}-10x}{-1}=-\frac{200}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{10}{-1}\right)x=-\frac{200}{-1}$$

Divide -10 by -1.

$$x^{2}+10x=-\frac{200}{-1}$$

Divide -200 by -1.

$$x^{2}+10x=200$$

Divide 10, the coefficient of the x term, by 2 to get 5. Then add the square of 5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+10x+5^{2}=200+5^{2}$$

Square 5.

$$x^{2}+10x+25=200+25$$

Add 200 to 25.

$$x^{2}+10x+25=225$$

Factor x^{2}+10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+5\right)^{2}=225$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+5\right)^{2}}=\sqrt{225}$$

Simplify.

$$x+5=15 x+5=-15$$

Subtract 5 from both sides of the equation.

$$x=10 x=-20$$