Solve x+7/x+4=x+3/x-1+4

$$\frac { x + 7 } { x + 4 } = \frac { x + 3 } { x - 1 } + 4$$

Solve for x

$$x=-\frac{1}{4}=-0.25$$

$$x=-3$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -4,1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(x+4\right), the least common multiple of x+4,x-1.

$$\left(x-1\right)\left(x+7\right)=\left(x+4\right)\left(x+3\right)+\left(x-1\right)\left(x+4\right)\times 4$$

Use the distributive property to multiply x-1 by x+7 and combine like terms.

$$x^{2}+6x-7=\left(x+4\right)\left(x+3\right)+\left(x-1\right)\left(x+4\right)\times 4$$

Use the distributive property to multiply x+4 by x+3 and combine like terms.

$$x^{2}+6x-7=x^{2}+7x+12+\left(x-1\right)\left(x+4\right)\times 4$$

Use the distributive property to multiply x-1 by x+4 and combine like terms.

$$x^{2}+6x-7=x^{2}+7x+12+\left(x^{2}+3x-4\right)\times 4$$

Use the distributive property to multiply x^{2}+3x-4 by 4.

$$x^{2}+6x-7=x^{2}+7x+12+4x^{2}+12x-16$$

Combine x^{2} and 4x^{2} to get 5x^{2}.

$$x^{2}+6x-7=5x^{2}+7x+12+12x-16$$

Combine 7x and 12x to get 19x.

$$x^{2}+6x-7=5x^{2}+19x+12-16$$

Subtract 16 from 12 to get -4.

$$x^{2}+6x-7=5x^{2}+19x-4$$

Subtract 5x^{2} from both sides.

$$x^{2}+6x-7-5x^{2}=19x-4$$

Combine x^{2} and -5x^{2} to get -4x^{2}.

$$-4x^{2}+6x-7=19x-4$$

Subtract 19x from both sides.

$$-4x^{2}+6x-7-19x=-4$$

Combine 6x and -19x to get -13x.

$$-4x^{2}-13x-7=-4$$

Add 4 to both sides.

$$-4x^{2}-13x-7+4=0$$

Add -7 and 4 to get -3.

$$-4x^{2}-13x-3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, -13 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\left(-4\right)\left(-3\right)}}{2\left(-4\right)}$$

Square -13.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\left(-4\right)\left(-3\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-\left(-13\right)±\sqrt{169+16\left(-3\right)}}{2\left(-4\right)}$$

Multiply 16 times -3.

$$x=\frac{-\left(-13\right)±\sqrt{169-48}}{2\left(-4\right)}$$

Add 169 to -48.

$$x=\frac{-\left(-13\right)±\sqrt{121}}{2\left(-4\right)}$$

Take the square root of 121.

$$x=\frac{-\left(-13\right)±11}{2\left(-4\right)}$$

The opposite of -13 is 13.

$$x=\frac{13±11}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{13±11}{-8}$$

Now solve the equation x=\frac{13±11}{-8} when ± is plus. Add 13 to 11.

$$x=\frac{24}{-8}$$

Divide 24 by -8.

$$x=-3$$

Now solve the equation x=\frac{13±11}{-8} when ± is minus. Subtract 11 from 13.

$$x=\frac{2}{-8}$$

Reduce the fraction \frac{2}{-8} to lowest terms by extracting and canceling out 2.

$$x=-\frac{1}{4}$$

The equation is now solved.

$$x=-3 x=-\frac{1}{4}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -4,1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(x+4\right), the least common multiple of x+4,x-1.

$$\left(x-1\right)\left(x+7\right)=\left(x+4\right)\left(x+3\right)+\left(x-1\right)\left(x+4\right)\times 4$$

Use the distributive property to multiply x-1 by x+7 and combine like terms.

$$x^{2}+6x-7=\left(x+4\right)\left(x+3\right)+\left(x-1\right)\left(x+4\right)\times 4$$

Use the distributive property to multiply x+4 by x+3 and combine like terms.

$$x^{2}+6x-7=x^{2}+7x+12+\left(x-1\right)\left(x+4\right)\times 4$$

Use the distributive property to multiply x-1 by x+4 and combine like terms.

$$x^{2}+6x-7=x^{2}+7x+12+\left(x^{2}+3x-4\right)\times 4$$

Use the distributive property to multiply x^{2}+3x-4 by 4.

$$x^{2}+6x-7=x^{2}+7x+12+4x^{2}+12x-16$$

Combine x^{2} and 4x^{2} to get 5x^{2}.

$$x^{2}+6x-7=5x^{2}+7x+12+12x-16$$

Combine 7x and 12x to get 19x.

$$x^{2}+6x-7=5x^{2}+19x+12-16$$

Subtract 16 from 12 to get -4.

$$x^{2}+6x-7=5x^{2}+19x-4$$

Subtract 5x^{2} from both sides.

$$x^{2}+6x-7-5x^{2}=19x-4$$

Combine x^{2} and -5x^{2} to get -4x^{2}.

$$-4x^{2}+6x-7=19x-4$$

Subtract 19x from both sides.

$$-4x^{2}+6x-7-19x=-4$$

Combine 6x and -19x to get -13x.

$$-4x^{2}-13x-7=-4$$

Add 7 to both sides.

$$-4x^{2}-13x=-4+7$$

Add -4 and 7 to get 3.

$$-4x^{2}-13x=3$$

Divide both sides by -4.

$$\frac{-4x^{2}-13x}{-4}=\frac{3}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\left(-\frac{13}{-4}\right)x=\frac{3}{-4}$$

Divide -13 by -4.

$$x^{2}+\frac{13}{4}x=\frac{3}{-4}$$

Divide 3 by -4.

$$x^{2}+\frac{13}{4}x=-\frac{3}{4}$$

Divide \frac{13}{4}, the coefficient of the x term, by 2 to get \frac{13}{8}. Then add the square of \frac{13}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{13}{4}x+\left(\frac{13}{8}\right)^{2}=-\frac{3}{4}+\left(\frac{13}{8}\right)^{2}$$

Square \frac{13}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{13}{4}x+\frac{169}{64}=-\frac{3}{4}+\frac{169}{64}$$

Add -\frac{3}{4} to \frac{169}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{13}{4}x+\frac{169}{64}=\frac{121}{64}$$

Factor x^{2}+\frac{13}{4}x+\frac{169}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{13}{8}\right)^{2}=\frac{121}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{13}{8}\right)^{2}}=\sqrt{\frac{121}{64}}$$

Simplify.

$$x+\frac{13}{8}=\frac{11}{8} x+\frac{13}{8}=-\frac{11}{8}$$

Subtract \frac{13}{8} from both sides of the equation.

$$x=-\frac{1}{4} x=-3$$