Solve 7y^2/x+4+7y+5/x+4=0

$$\frac { 7 y ^ { 2 } } { x + 4 } + \frac { 7 y + 5 } { x + 4 } = 0$$

Solve for y

$$y=\frac{\sqrt{91}i}{14}-\frac{1}{2}$$

$$y=-\frac{\sqrt{91}i}{14}-\frac{1}{2}\text{, }x\neq -4$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by x+4.

$$7y^{2}+7y+5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, 7 for b, and 5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-7±\sqrt{7^{2}-4\times 7\times 5}}{2\times 7}$$

Square 7.

$$y=\frac{-7±\sqrt{49-4\times 7\times 5}}{2\times 7}$$

Multiply -4 times 7.

$$y=\frac{-7±\sqrt{49-28\times 5}}{2\times 7}$$

Multiply -28 times 5.

$$y=\frac{-7±\sqrt{49-140}}{2\times 7}$$

Add 49 to -140.

$$y=\frac{-7±\sqrt{-91}}{2\times 7}$$

Take the square root of -91.

$$y=\frac{-7±\sqrt{91}i}{2\times 7}$$

Multiply 2 times 7.

$$y=\frac{-7±\sqrt{91}i}{14}$$

Now solve the equation y=\frac{-7±\sqrt{91}i}{14} when ± is plus. Add -7 to i\sqrt{91}.

$$y=\frac{-7+\sqrt{91}i}{14}$$

Divide -7+i\sqrt{91} by 14.

$$y=\frac{\sqrt{91}i}{14}-\frac{1}{2}$$

Now solve the equation y=\frac{-7±\sqrt{91}i}{14} when ± is minus. Subtract i\sqrt{91} from -7.

$$y=\frac{-\sqrt{91}i-7}{14}$$

Divide -7-i\sqrt{91} by 14.

$$y=-\frac{\sqrt{91}i}{14}-\frac{1}{2}$$

The equation is now solved.

$$y=\frac{\sqrt{91}i}{14}-\frac{1}{2} y=-\frac{\sqrt{91}i}{14}-\frac{1}{2}$$

Steps for Completing the Square

Multiply both sides of the equation by x+4.

$$7y^{2}+7y+5=0$$

Subtract 5 from both sides. Anything subtracted from zero gives its negation.

$$7y^{2}+7y=-5$$

Divide both sides by 7.

$$\frac{7y^{2}+7y}{7}=-\frac{5}{7}$$

Dividing by 7 undoes the multiplication by 7.

$$y^{2}+\frac{7}{7}y=-\frac{5}{7}$$

Divide 7 by 7.

$$y^{2}+y=-\frac{5}{7}$$

Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+y+\left(\frac{1}{2}\right)^{2}=-\frac{5}{7}+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$y^{2}+y+\frac{1}{4}=-\frac{5}{7}+\frac{1}{4}$$

Add -\frac{5}{7} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}+y+\frac{1}{4}=-\frac{13}{28}$$

Factor y^{2}+y+\frac{1}{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}.

$$\left(y+\frac{1}{2}\right)^{2}=-\frac{13}{28}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+\frac{1}{2}\right)^{2}}=\sqrt{-\frac{13}{28}}$$

Simplify.

$$y+\frac{1}{2}=\frac{\sqrt{91}i}{14} y+\frac{1}{2}=-\frac{\sqrt{91}i}{14}$$

Subtract \frac{1}{2} from both sides of the equation.

$$y=\frac{\sqrt{91}i}{14}-\frac{1}{2} y=-\frac{\sqrt{91}i}{14}-\frac{1}{2}$$

Solve for x

$$x\neq -4$$

$$y=\frac{\sqrt{91}i}{14}-\frac{1}{2}\text{ or }y=-\frac{\sqrt{91}i}{14}-\frac{1}{2}$$