Solve x/x+1=2x/5x+3+1

$$\frac { x } { x + 1 } = \frac { 2 x } { 5 x + 3 } + 1$$

Solve for x

$$x=-\frac{1}{2}=-0.5$$

$$x=-3$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -1,-\frac{3}{5} since division by zero is not defined. Multiply both sides of the equation by \left(x+1\right)\left(5x+3\right), the least common multiple of x+1,5x+3.

$$\left(5x+3\right)x=\left(x+1\right)\times 2x+\left(x+1\right)\left(5x+3\right)$$

Use the distributive property to multiply 5x+3 by x.

$$5x^{2}+3x=\left(x+1\right)\times 2x+\left(x+1\right)\left(5x+3\right)$$

Use the distributive property to multiply x+1 by 2.

$$5x^{2}+3x=\left(2x+2\right)x+\left(x+1\right)\left(5x+3\right)$$

Use the distributive property to multiply 2x+2 by x.

$$5x^{2}+3x=2x^{2}+2x+\left(x+1\right)\left(5x+3\right)$$

Use the distributive property to multiply x+1 by 5x+3 and combine like terms.

$$5x^{2}+3x=2x^{2}+2x+5x^{2}+8x+3$$

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

$$5x^{2}+3x=7x^{2}+2x+8x+3$$

Combine 2x and 8x to get 10x.

$$5x^{2}+3x=7x^{2}+10x+3$$

Subtract 7x^{2} from both sides.

$$5x^{2}+3x-7x^{2}=10x+3$$

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

$$-2x^{2}+3x=10x+3$$

Subtract 10x from both sides.

$$-2x^{2}+3x-10x=3$$

Combine 3x and -10x to get -7x.

$$-2x^{2}-7x=3$$

Subtract 3 from both sides.

$$-2x^{2}-7x-3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, -7 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-2\right)\left(-3\right)}}{2\left(-2\right)}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\left(-2\right)\left(-3\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-\left(-7\right)±\sqrt{49+8\left(-3\right)}}{2\left(-2\right)}$$

Multiply 8 times -3.

$$x=\frac{-\left(-7\right)±\sqrt{49-24}}{2\left(-2\right)}$$

Add 49 to -24.

$$x=\frac{-\left(-7\right)±\sqrt{25}}{2\left(-2\right)}$$

Take the square root of 25.

$$x=\frac{-\left(-7\right)±5}{2\left(-2\right)}$$

The opposite of -7 is 7.

$$x=\frac{7±5}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{7±5}{-4}$$

Now solve the equation x=\frac{7±5}{-4} when ± is plus. Add 7 to 5.

$$x=\frac{12}{-4}$$

Divide 12 by -4.

$$x=-3$$

Now solve the equation x=\frac{7±5}{-4} when ± is minus. Subtract 5 from 7.

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

Reduce the fraction \frac{2}{-4} to lowest terms by extracting and canceling out 2.

$$x=-\frac{1}{2}$$

The equation is now solved.

$$x=-3 x=-\frac{1}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -1,-\frac{3}{5} since division by zero is not defined. Multiply both sides of the equation by \left(x+1\right)\left(5x+3\right), the least common multiple of x+1,5x+3.

$$\left(5x+3\right)x=\left(x+1\right)\times 2x+\left(x+1\right)\left(5x+3\right)$$

Use the distributive property to multiply 5x+3 by x.

$$5x^{2}+3x=\left(x+1\right)\times 2x+\left(x+1\right)\left(5x+3\right)$$

Use the distributive property to multiply x+1 by 2.

$$5x^{2}+3x=\left(2x+2\right)x+\left(x+1\right)\left(5x+3\right)$$

Use the distributive property to multiply 2x+2 by x.

$$5x^{2}+3x=2x^{2}+2x+\left(x+1\right)\left(5x+3\right)$$

Use the distributive property to multiply x+1 by 5x+3 and combine like terms.

$$5x^{2}+3x=2x^{2}+2x+5x^{2}+8x+3$$

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

$$5x^{2}+3x=7x^{2}+2x+8x+3$$

Combine 2x and 8x to get 10x.

$$5x^{2}+3x=7x^{2}+10x+3$$

Subtract 7x^{2} from both sides.

$$5x^{2}+3x-7x^{2}=10x+3$$

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

$$-2x^{2}+3x=10x+3$$

Subtract 10x from both sides.

$$-2x^{2}+3x-10x=3$$

Combine 3x and -10x to get -7x.

$$-2x^{2}-7x=3$$

Divide both sides by -2.

$$\frac{-2x^{2}-7x}{-2}=\frac{3}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\left(-\frac{7}{-2}\right)x=\frac{3}{-2}$$

Divide -7 by -2.

$$x^{2}+\frac{7}{2}x=\frac{3}{-2}$$

Divide 3 by -2.

$$x^{2}+\frac{7}{2}x=-\frac{3}{2}$$

Divide \frac{7}{2}, the coefficient of the x term, by 2 to get \frac{7}{4}. Then add the square of \frac{7}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{7}{2}x+\left(\frac{7}{4}\right)^{2}=-\frac{3}{2}+\left(\frac{7}{4}\right)^{2}$$

Square \frac{7}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{7}{2}x+\frac{49}{16}=-\frac{3}{2}+\frac{49}{16}$$

Add -\frac{3}{2} to \frac{49}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{7}{2}x+\frac{49}{16}=\frac{25}{16}$$

Factor x^{2}+\frac{7}{2}x+\frac{49}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{7}{4}\right)^{2}=\frac{25}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{4}\right)^{2}}=\sqrt{\frac{25}{16}}$$

Simplify.

$$x+\frac{7}{4}=\frac{5}{4} x+\frac{7}{4}=-\frac{5}{4}$$

Subtract \frac{7}{4} from both sides of the equation.

$$x=-\frac{1}{2} x=-3$$