Solve -6x+6/x-7+1/x-1=-8/x^2-8x+7

$$\frac { - 6 x + 6 } { x - 7 } + \frac { 1 } { x - 1 } = - \frac { 8 } { x ^ { 2 } - 8 x + 7 }$$

Solve for x

$$x = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

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

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values 1,7 since division by zero is not defined. Multiply both sides of the equation by \left(x-7\right)\left(x-1\right), the least common multiple of x-7,x-1,x^{2}-8x+7.

$$\left(x-1\right)\left(-6x+6\right)+x-7=-8$$

Use the distributive property to multiply x-1 by -6x+6 and combine like terms.

$$-6x^{2}+12x-6+x-7=-8$$

Combine 12x and x to get 13x.

$$-6x^{2}+13x-6-7=-8$$

Subtract 7 from -6 to get -13.

$$-6x^{2}+13x-13=-8$$

Add 8 to both sides.

$$-6x^{2}+13x-13+8=0$$

Add -13 and 8 to get -5.

$$-6x^{2}+13x-5=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -6x^{2}+ax+bx-5. To find a and b, set up a system to be solved.

$$a+b=13 ab=-6\left(-5\right)=30$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 30.

$$1,30 2,15 3,10 5,6$$

Calculate the sum for each pair.

$$1+30=31 2+15=17 3+10=13 5+6=11$$

The solution is the pair that gives sum 13.

$$a=10 b=3$$

Rewrite -6x^{2}+13x-5 as \left(-6x^{2}+10x\right)+\left(3x-5\right).

$$\left(-6x^{2}+10x\right)+\left(3x-5\right)$$

Factor out 2x in the first and -1 in the second group.

$$2x\left(-3x+5\right)-\left(-3x+5\right)$$

Factor out common term -3x+5 by using distributive property.

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

To find equation solutions, solve -3x+5=0 and 2x-1=0.

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

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 1,7 since division by zero is not defined. Multiply both sides of the equation by \left(x-7\right)\left(x-1\right), the least common multiple of x-7,x-1,x^{2}-8x+7.

$$\left(x-1\right)\left(-6x+6\right)+x-7=-8$$

Use the distributive property to multiply x-1 by -6x+6 and combine like terms.

$$-6x^{2}+12x-6+x-7=-8$$

Combine 12x and x to get 13x.

$$-6x^{2}+13x-6-7=-8$$

Subtract 7 from -6 to get -13.

$$-6x^{2}+13x-13=-8$$

Add 8 to both sides.

$$-6x^{2}+13x-13+8=0$$

Add -13 and 8 to get -5.

$$-6x^{2}+13x-5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6 for a, 13 for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}-4\left(-6\right)\left(-5\right)}}{2\left(-6\right)}$$

Square 13.

$$x=\frac{-13±\sqrt{169-4\left(-6\right)\left(-5\right)}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{-13±\sqrt{169+24\left(-5\right)}}{2\left(-6\right)}$$

Multiply 24 times -5.

$$x=\frac{-13±\sqrt{169-120}}{2\left(-6\right)}$$

Add 169 to -120.

$$x=\frac{-13±\sqrt{49}}{2\left(-6\right)}$$

Take the square root of 49.

$$x=\frac{-13±7}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{-13±7}{-12}$$

Now solve the equation x=\frac{-13±7}{-12} when ± is plus. Add -13 to 7.

$$x=-\frac{6}{-12}$$

Reduce the fraction \frac{-6}{-12} to lowest terms by extracting and canceling out 6.

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

Now solve the equation x=\frac{-13±7}{-12} when ± is minus. Subtract 7 from -13.

$$x=-\frac{20}{-12}$$

Reduce the fraction \frac{-20}{-12} to lowest terms by extracting and canceling out 4.

$$x=\frac{5}{3}$$

The equation is now solved.

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

Steps for Completing the Square

Variable x cannot be equal to any of the values 1,7 since division by zero is not defined. Multiply both sides of the equation by \left(x-7\right)\left(x-1\right), the least common multiple of x-7,x-1,x^{2}-8x+7.

$$\left(x-1\right)\left(-6x+6\right)+x-7=-8$$

Use the distributive property to multiply x-1 by -6x+6 and combine like terms.

$$-6x^{2}+12x-6+x-7=-8$$

Combine 12x and x to get 13x.

$$-6x^{2}+13x-6-7=-8$$

Subtract 7 from -6 to get -13.

$$-6x^{2}+13x-13=-8$$

Add 13 to both sides.

$$-6x^{2}+13x=-8+13$$

Add -8 and 13 to get 5.

$$-6x^{2}+13x=5$$

Divide both sides by -6.

$$\frac{-6x^{2}+13x}{-6}=\frac{5}{-6}$$

Dividing by -6 undoes the multiplication by -6.

$$x^{2}+\frac{13}{-6}x=\frac{5}{-6}$$

Divide 13 by -6.

$$x^{2}-\frac{13}{6}x=\frac{5}{-6}$$

Divide 5 by -6.

$$x^{2}-\frac{13}{6}x=-\frac{5}{6}$$

Divide -\frac{13}{6}, the coefficient of the x term, by 2 to get -\frac{13}{12}. Then add the square of -\frac{13}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{13}{6}x+\left(-\frac{13}{12}\right)^{2}=-\frac{5}{6}+\left(-\frac{13}{12}\right)^{2}$$

Square -\frac{13}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{13}{6}x+\frac{169}{144}=-\frac{5}{6}+\frac{169}{144}$$

Add -\frac{5}{6} to \frac{169}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{13}{6}x+\frac{169}{144}=\frac{49}{144}$$

Factor x^{2}-\frac{13}{6}x+\frac{169}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{13}{12}\right)^{2}=\frac{49}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{12}\right)^{2}}=\sqrt{\frac{49}{144}}$$

Simplify.

$$x-\frac{13}{12}=\frac{7}{12} x-\frac{13}{12}=-\frac{7}{12}$$

Add \frac{13}{12} to both sides of the equation.

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