Solve x+8/5x+7=x+8/7x+5

$$\frac{ x+8 }{ 5x+7 } = \frac{ x+8 }{ 7x+5 }$$

Solve for x

$$x=1$$

$$x=-8$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -\frac{7}{5},-\frac{5}{7} since division by zero is not defined. Multiply both sides of the equation by \left(5x+7\right)\left(7x+5\right), the least common multiple of 5x+7,7x+5.

$$\left(7x+5\right)\left(x+8\right)=\left(5x+7\right)\left(x+8\right)$$

Use the distributive property to multiply 7x+5 by x+8 and combine like terms.

$$7x^{2}+61x+40=\left(5x+7\right)\left(x+8\right)$$

Use the distributive property to multiply 5x+7 by x+8 and combine like terms.

$$7x^{2}+61x+40=5x^{2}+47x+56$$

Subtract 5x^{2} from both sides.

$$7x^{2}+61x+40-5x^{2}=47x+56$$

Combine 7x^{2} and -5x^{2} to get 2x^{2}.

$$2x^{2}+61x+40=47x+56$$

Subtract 47x from both sides.

$$2x^{2}+61x+40-47x=56$$

Combine 61x and -47x to get 14x.

$$2x^{2}+14x+40=56$$

Subtract 56 from both sides.

$$2x^{2}+14x+40-56=0$$

Subtract 56 from 40 to get -16.

$$2x^{2}+14x-16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 14 for b, and -16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\times 2\left(-16\right)}}{2\times 2}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\times 2\left(-16\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-14±\sqrt{196-8\left(-16\right)}}{2\times 2}$$

Multiply -8 times -16.

$$x=\frac{-14±\sqrt{196+128}}{2\times 2}$$

Add 196 to 128.

$$x=\frac{-14±\sqrt{324}}{2\times 2}$$

Take the square root of 324.

$$x=\frac{-14±18}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-14±18}{4}$$

Now solve the equation x=\frac{-14±18}{4} when ± is plus. Add -14 to 18.

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

Divide 4 by 4.

$$x=1$$

Now solve the equation x=\frac{-14±18}{4} when ± is minus. Subtract 18 from -14.

$$x=-\frac{32}{4}$$

Divide -32 by 4.

$$x=-8$$

The equation is now solved.

$$x=1 x=-8$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -\frac{7}{5},-\frac{5}{7} since division by zero is not defined. Multiply both sides of the equation by \left(5x+7\right)\left(7x+5\right), the least common multiple of 5x+7,7x+5.

$$\left(7x+5\right)\left(x+8\right)=\left(5x+7\right)\left(x+8\right)$$

Use the distributive property to multiply 7x+5 by x+8 and combine like terms.

$$7x^{2}+61x+40=\left(5x+7\right)\left(x+8\right)$$

Use the distributive property to multiply 5x+7 by x+8 and combine like terms.

$$7x^{2}+61x+40=5x^{2}+47x+56$$

Subtract 5x^{2} from both sides.

$$7x^{2}+61x+40-5x^{2}=47x+56$$

Combine 7x^{2} and -5x^{2} to get 2x^{2}.

$$2x^{2}+61x+40=47x+56$$

Subtract 47x from both sides.

$$2x^{2}+61x+40-47x=56$$

Combine 61x and -47x to get 14x.

$$2x^{2}+14x+40=56$$

Subtract 40 from both sides.

$$2x^{2}+14x=56-40$$

Subtract 40 from 56 to get 16.

$$2x^{2}+14x=16$$

Divide both sides by 2.

$$\frac{2x^{2}+14x}{2}=\frac{16}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{14}{2}x=\frac{16}{2}$$

Divide 14 by 2.

$$x^{2}+7x=\frac{16}{2}$$

Divide 16 by 2.

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

Divide 7, the coefficient of the x term, by 2 to get \frac{7}{2}. Then add the square of \frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+7x+\left(\frac{7}{2}\right)^{2}=8+\left(\frac{7}{2}\right)^{2}$$

Square \frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+7x+\frac{49}{4}=8+\frac{49}{4}$$

Add 8 to \frac{49}{4}.

$$x^{2}+7x+\frac{49}{4}=\frac{81}{4}$$

Factor x^{2}+7x+\frac{49}{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}.

$$\left(x+\frac{7}{2}\right)^{2}=\frac{81}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{2}\right)^{2}}=\sqrt{\frac{81}{4}}$$

Simplify.

$$x+\frac{7}{2}=\frac{9}{2} x+\frac{7}{2}=-\frac{9}{2}$$

Subtract \frac{7}{2} from both sides of the equation.

$$x=1 x=-8$$