Solve 3x-7/x-5+x/2=8/x-5

$$\frac{ 3x-7 }{ x-5 } + \frac{ x }{ 2 } = \frac{ 8 }{ x-5 }$$

Solve for x

$$x=-6$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 5 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-5\right), the least common multiple of x-5,2.

$$2\left(3x-7\right)+\left(x-5\right)x=2\times 8$$

Use the distributive property to multiply 2 by 3x-7.

$$6x-14+\left(x-5\right)x=2\times 8$$

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

$$6x-14+x^{2}-5x=2\times 8$$

Combine 6x and -5x to get x.

$$x-14+x^{2}=2\times 8$$

Multiply 2 and 8 to get 16.

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

Subtract 16 from both sides.

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

Subtract 16 from -14 to get -30.

$$x-30+x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}+x-30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 1 for b, and -30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-30\right)}}{2}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\left(-30\right)}}{2}$$

Multiply -4 times -30.

$$x=\frac{-1±\sqrt{1+120}}{2}$$

Add 1 to 120.

$$x=\frac{-1±\sqrt{121}}{2}$$

Take the square root of 121.

$$x=\frac{-1±11}{2}$$

Now solve the equation x=\frac{-1±11}{2} when ± is plus. Add -1 to 11.

$$x=\frac{10}{2}$$

Divide 10 by 2.

$$x=5$$

Now solve the equation x=\frac{-1±11}{2} when ± is minus. Subtract 11 from -1.

$$x=-\frac{12}{2}$$

Divide -12 by 2.

$$x=-6$$

The equation is now solved.

$$x=5 x=-6$$

Variable x cannot be equal to 5.

$$x=-6$$

Steps for Completing the Square

Variable x cannot be equal to 5 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-5\right), the least common multiple of x-5,2.

$$2\left(3x-7\right)+\left(x-5\right)x=2\times 8$$

Use the distributive property to multiply 2 by 3x-7.

$$6x-14+\left(x-5\right)x=2\times 8$$

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

$$6x-14+x^{2}-5x=2\times 8$$

Combine 6x and -5x to get x.

$$x-14+x^{2}=2\times 8$$

Multiply 2 and 8 to get 16.

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

Add 14 to both sides.

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

Add 16 and 14 to get 30.

$$x+x^{2}=30$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}+x=30$$

Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=30+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+x+\frac{1}{4}=30+\frac{1}{4}$$

Add 30 to \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=\frac{121}{4}$$

Factor x^{2}+x+\frac{1}{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}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{121}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Simplify.

$$x+\frac{1}{2}=\frac{11}{2} x+\frac{1}{2}=-\frac{11}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=5 x=-6$$

Variable x cannot be equal to 5.

$$x=-6$$