Solve 5x-8/x-1=7x-4/x+2

$$\frac{ 5x-8 }{ x-1 } = \frac{ 7x-4 }{ x+2 }$$

Solve for x

$$x=4$$

$$x = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

Steps Using the Quadratic Formula

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

$$\left(x+2\right)\left(5x-8\right)=\left(x-1\right)\left(7x-4\right)$$

Use the distributive property to multiply x+2 by 5x-8 and combine like terms.

$$5x^{2}+2x-16=\left(x-1\right)\left(7x-4\right)$$

Use the distributive property to multiply x-1 by 7x-4 and combine like terms.

$$5x^{2}+2x-16=7x^{2}-11x+4$$

Subtract 7x^{2} from both sides.

$$5x^{2}+2x-16-7x^{2}=-11x+4$$

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

$$-2x^{2}+2x-16=-11x+4$$

Add 11x to both sides.

$$-2x^{2}+2x-16+11x=4$$

Combine 2x and 11x to get 13x.

$$-2x^{2}+13x-16=4$$

Subtract 4 from both sides.

$$-2x^{2}+13x-16-4=0$$

Subtract 4 from -16 to get -20.

$$-2x^{2}+13x-20=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 13 for b, and -20 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}-4\left(-2\right)\left(-20\right)}}{2\left(-2\right)}$$

Square 13.

$$x=\frac{-13±\sqrt{169-4\left(-2\right)\left(-20\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-13±\sqrt{169+8\left(-20\right)}}{2\left(-2\right)}$$

Multiply 8 times -20.

$$x=\frac{-13±\sqrt{169-160}}{2\left(-2\right)}$$

Add 169 to -160.

$$x=\frac{-13±\sqrt{9}}{2\left(-2\right)}$$

Take the square root of 9.

$$x=\frac{-13±3}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-13±3}{-4}$$

Now solve the equation x=\frac{-13±3}{-4} when ± is plus. Add -13 to 3.

$$x=-\frac{10}{-4}$$

Reduce the fraction \frac{-10}{-4} to lowest terms by extracting and canceling out 2.

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

Now solve the equation x=\frac{-13±3}{-4} when ± is minus. Subtract 3 from -13.

$$x=-\frac{16}{-4}$$

Divide -16 by -4.

$$x=4$$

The equation is now solved.

$$x=\frac{5}{2} x=4$$

Steps for Completing the Square

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

$$\left(x+2\right)\left(5x-8\right)=\left(x-1\right)\left(7x-4\right)$$

Use the distributive property to multiply x+2 by 5x-8 and combine like terms.

$$5x^{2}+2x-16=\left(x-1\right)\left(7x-4\right)$$

Use the distributive property to multiply x-1 by 7x-4 and combine like terms.

$$5x^{2}+2x-16=7x^{2}-11x+4$$

Subtract 7x^{2} from both sides.

$$5x^{2}+2x-16-7x^{2}=-11x+4$$

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

$$-2x^{2}+2x-16=-11x+4$$

Add 11x to both sides.

$$-2x^{2}+2x-16+11x=4$$

Combine 2x and 11x to get 13x.

$$-2x^{2}+13x-16=4$$

Add 16 to both sides.

$$-2x^{2}+13x=4+16$$

Add 4 and 16 to get 20.

$$-2x^{2}+13x=20$$

Divide both sides by -2.

$$\frac{-2x^{2}+13x}{-2}=\frac{20}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{13}{-2}x=\frac{20}{-2}$$

Divide 13 by -2.

$$x^{2}-\frac{13}{2}x=\frac{20}{-2}$$

Divide 20 by -2.

$$x^{2}-\frac{13}{2}x=-10$$

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

$$x^{2}-\frac{13}{2}x+\left(-\frac{13}{4}\right)^{2}=-10+\left(-\frac{13}{4}\right)^{2}$$

Square -\frac{13}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{13}{2}x+\frac{169}{16}=-10+\frac{169}{16}$$

Add -10 to \frac{169}{16}.

$$x^{2}-\frac{13}{2}x+\frac{169}{16}=\frac{9}{16}$$

Factor x^{2}-\frac{13}{2}x+\frac{169}{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}.

$$\left(x-\frac{13}{4}\right)^{2}=\frac{9}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{4}\right)^{2}}=\sqrt{\frac{9}{16}}$$

Simplify.

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

Add \frac{13}{4} to both sides of the equation.

$$x=4 x=\frac{5}{2}$$