Solve 4x+19/x+1=x+7

$$\frac { 4 x + 19 } { x + 1 } = x + 7$$

Solve for x

$$x=-6$$

$$x=2$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by x+1.

$$4x+19=\left(x+1\right)x+\left(x+1\right)\times 7$$

Use the distributive property to multiply x+1 by x.

$$4x+19=x^{2}+x+\left(x+1\right)\times 7$$

Use the distributive property to multiply x+1 by 7.

$$4x+19=x^{2}+x+7x+7$$

Combine x and 7x to get 8x.

$$4x+19=x^{2}+8x+7$$

Subtract x^{2} from both sides.

$$4x+19-x^{2}=8x+7$$

Subtract 8x from both sides.

$$4x+19-x^{2}-8x=7$$

Combine 4x and -8x to get -4x.

$$-4x+19-x^{2}=7$$

Subtract 7 from both sides.

$$-4x+19-x^{2}-7=0$$

Subtract 7 from 19 to get 12.

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

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.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -4 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-1\right)\times 12}}{2\left(-1\right)}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-1\right)\times 12}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-4\right)±\sqrt{16+4\times 12}}{2\left(-1\right)}$$

Multiply 4 times 12.

$$x=\frac{-\left(-4\right)±\sqrt{16+48}}{2\left(-1\right)}$$

Add 16 to 48.

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

Take the square root of 64.

$$x=\frac{-\left(-4\right)±8}{2\left(-1\right)}$$

The opposite of -4 is 4.

$$x=\frac{4±8}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{4±8}{-2}$$

Now solve the equation x=\frac{4±8}{-2} when ± is plus. Add 4 to 8.

$$x=\frac{12}{-2}$$

Divide 12 by -2.

$$x=-6$$

Now solve the equation x=\frac{4±8}{-2} when ± is minus. Subtract 8 from 4.

$$x=-\frac{4}{-2}$$

Divide -4 by -2.

$$x=2$$

The equation is now solved.

$$x=-6 x=2$$

Steps for Completing the Square

Variable x cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by x+1.

$$4x+19=\left(x+1\right)x+\left(x+1\right)\times 7$$

Use the distributive property to multiply x+1 by x.

$$4x+19=x^{2}+x+\left(x+1\right)\times 7$$

Use the distributive property to multiply x+1 by 7.

$$4x+19=x^{2}+x+7x+7$$

Combine x and 7x to get 8x.

$$4x+19=x^{2}+8x+7$$

Subtract x^{2} from both sides.

$$4x+19-x^{2}=8x+7$$

Subtract 8x from both sides.

$$4x+19-x^{2}-8x=7$$

Combine 4x and -8x to get -4x.

$$-4x+19-x^{2}=7$$

Subtract 19 from both sides.

$$-4x-x^{2}=7-19$$

Subtract 19 from 7 to get -12.

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

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.

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

Divide both sides by -1.

$$\frac{-x^{2}-4x}{-1}=-\frac{12}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{4}{-1}\right)x=-\frac{12}{-1}$$

Divide -4 by -1.

$$x^{2}+4x=-\frac{12}{-1}$$

Divide -12 by -1.

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

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

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

Square 2.

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

Add 12 to 4.

$$x^{2}+4x+4=16$$

Factor x^{2}+4x+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}.

$$\left(x+2\right)^{2}=16$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{16}$$

Simplify.

$$x+2=4 x+2=-4$$

Subtract 2 from both sides of the equation.

$$x=2 x=-6$$