Solve x^2+6x-7/3x^2-x-2=5

$$\frac { x ^ { 2 } + 6 x - 7 } { 3 x ^ { 2 } - x - 2 } = 5$$

Solve for x

$$x=-\frac{3}{14}\approx -0.214285714$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -\frac{2}{3},1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(3x+2\right).

$$x^{2}+6x-7=5\left(x-1\right)\left(3x+2\right)$$

Use the distributive property to multiply 5 by x-1.

$$x^{2}+6x-7=\left(5x-5\right)\left(3x+2\right)$$

Use the distributive property to multiply 5x-5 by 3x+2 and combine like terms.

$$x^{2}+6x-7=15x^{2}-5x-10$$

Subtract 15x^{2} from both sides.

$$x^{2}+6x-7-15x^{2}=-5x-10$$

Combine x^{2} and -15x^{2} to get -14x^{2}.

$$-14x^{2}+6x-7=-5x-10$$

Add 5x to both sides.

$$-14x^{2}+6x-7+5x=-10$$

Combine 6x and 5x to get 11x.

$$-14x^{2}+11x-7=-10$$

Add 10 to both sides.

$$-14x^{2}+11x-7+10=0$$

Add -7 and 10 to get 3.

$$-14x^{2}+11x+3=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -14x^{2}+ax+bx+3. To find a and b, set up a system to be solved.

$$a+b=11 ab=-14\times 3=-42$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -42.

$$-1,42 -2,21 -3,14 -6,7$$

Calculate the sum for each pair.

$$-1+42=41 -2+21=19 -3+14=11 -6+7=1$$

The solution is the pair that gives sum 11.

$$a=14 b=-3$$

Rewrite -14x^{2}+11x+3 as \left(-14x^{2}+14x\right)+\left(-3x+3\right).

$$\left(-14x^{2}+14x\right)+\left(-3x+3\right)$$

Factor out 14x in the first and 3 in the second group.

$$14x\left(-x+1\right)+3\left(-x+1\right)$$

Factor out common term -x+1 by using distributive property.

$$\left(-x+1\right)\left(14x+3\right)$$

To find equation solutions, solve -x+1=0 and 14x+3=0.

$$x=1 x=-\frac{3}{14}$$

Variable x cannot be equal to 1.

$$x=-\frac{3}{14}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -\frac{2}{3},1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(3x+2\right).

$$x^{2}+6x-7=5\left(x-1\right)\left(3x+2\right)$$

Use the distributive property to multiply 5 by x-1.

$$x^{2}+6x-7=\left(5x-5\right)\left(3x+2\right)$$

Use the distributive property to multiply 5x-5 by 3x+2 and combine like terms.

$$x^{2}+6x-7=15x^{2}-5x-10$$

Subtract 15x^{2} from both sides.

$$x^{2}+6x-7-15x^{2}=-5x-10$$

Combine x^{2} and -15x^{2} to get -14x^{2}.

$$-14x^{2}+6x-7=-5x-10$$

Add 5x to both sides.

$$-14x^{2}+6x-7+5x=-10$$

Combine 6x and 5x to get 11x.

$$-14x^{2}+11x-7=-10$$

Add 10 to both sides.

$$-14x^{2}+11x-7+10=0$$

Add -7 and 10 to get 3.

$$-14x^{2}+11x+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -14 for a, 11 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-11±\sqrt{11^{2}-4\left(-14\right)\times 3}}{2\left(-14\right)}$$

Square 11.

$$x=\frac{-11±\sqrt{121-4\left(-14\right)\times 3}}{2\left(-14\right)}$$

Multiply -4 times -14.

$$x=\frac{-11±\sqrt{121+56\times 3}}{2\left(-14\right)}$$

Multiply 56 times 3.

$$x=\frac{-11±\sqrt{121+168}}{2\left(-14\right)}$$

Add 121 to 168.

$$x=\frac{-11±\sqrt{289}}{2\left(-14\right)}$$

Take the square root of 289.

$$x=\frac{-11±17}{2\left(-14\right)}$$

Multiply 2 times -14.

$$x=\frac{-11±17}{-28}$$

Now solve the equation x=\frac{-11±17}{-28} when ± is plus. Add -11 to 17.

$$x=\frac{6}{-28}$$

Reduce the fraction \frac{6}{-28} to lowest terms by extracting and canceling out 2.

$$x=-\frac{3}{14}$$

Now solve the equation x=\frac{-11±17}{-28} when ± is minus. Subtract 17 from -11.

$$x=-\frac{28}{-28}$$

Divide -28 by -28.

$$x=1$$

The equation is now solved.

$$x=-\frac{3}{14} x=1$$

Variable x cannot be equal to 1.

$$x=-\frac{3}{14}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -\frac{2}{3},1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(3x+2\right).

$$x^{2}+6x-7=5\left(x-1\right)\left(3x+2\right)$$

Use the distributive property to multiply 5 by x-1.

$$x^{2}+6x-7=\left(5x-5\right)\left(3x+2\right)$$

Use the distributive property to multiply 5x-5 by 3x+2 and combine like terms.

$$x^{2}+6x-7=15x^{2}-5x-10$$

Subtract 15x^{2} from both sides.

$$x^{2}+6x-7-15x^{2}=-5x-10$$

Combine x^{2} and -15x^{2} to get -14x^{2}.

$$-14x^{2}+6x-7=-5x-10$$

Add 5x to both sides.

$$-14x^{2}+6x-7+5x=-10$$

Combine 6x and 5x to get 11x.

$$-14x^{2}+11x-7=-10$$

Add 7 to both sides.

$$-14x^{2}+11x=-10+7$$

Add -10 and 7 to get -3.

$$-14x^{2}+11x=-3$$

Divide both sides by -14.

$$\frac{-14x^{2}+11x}{-14}=-\frac{3}{-14}$$

Dividing by -14 undoes the multiplication by -14.

$$x^{2}+\frac{11}{-14}x=-\frac{3}{-14}$$

Divide 11 by -14.

$$x^{2}-\frac{11}{14}x=-\frac{3}{-14}$$

Divide -3 by -14.

$$x^{2}-\frac{11}{14}x=\frac{3}{14}$$

Divide -\frac{11}{14}, the coefficient of the x term, by 2 to get -\frac{11}{28}. Then add the square of -\frac{11}{28} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{11}{14}x+\left(-\frac{11}{28}\right)^{2}=\frac{3}{14}+\left(-\frac{11}{28}\right)^{2}$$

Square -\frac{11}{28} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{11}{14}x+\frac{121}{784}=\frac{3}{14}+\frac{121}{784}$$

Add \frac{3}{14} to \frac{121}{784} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{11}{14}x+\frac{121}{784}=\frac{289}{784}$$

Factor x^{2}-\frac{11}{14}x+\frac{121}{784}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{11}{28}\right)^{2}=\frac{289}{784}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{11}{28}\right)^{2}}=\sqrt{\frac{289}{784}}$$

Simplify.

$$x-\frac{11}{28}=\frac{17}{28} x-\frac{11}{28}=-\frac{17}{28}$$

Add \frac{11}{28} to both sides of the equation.

$$x=1 x=-\frac{3}{14}$$

Variable x cannot be equal to 1.

$$x=-\frac{3}{14}$$