Solve x+1/x-2=3x-7/x-1

$$\frac { x + 1 } { x - 2 } = \frac { 3 x - 7 } { x - 1 }$$

Solve for x

$$x=5$$

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

Steps Using the Quadratic Formula

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

$$\left(x-1\right)\left(x+1\right)=\left(x-2\right)\left(3x-7\right)$$

Consider \left(x-1\right)\left(x+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

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

Use the distributive property to multiply x-2 by 3x-7 and combine like terms.

$$x^{2}-1=3x^{2}-13x+14$$

Subtract 3x^{2} from both sides.

$$x^{2}-1-3x^{2}=-13x+14$$

Combine x^{2} and -3x^{2} to get -2x^{2}.

$$-2x^{2}-1=-13x+14$$

Add 13x to both sides.

$$-2x^{2}-1+13x=14$$

Subtract 14 from both sides.

$$-2x^{2}-1+13x-14=0$$

Subtract 14 from -1 to get -15.

$$-2x^{2}-15+13x=0$$

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.

$$-2x^{2}+13x-15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 13 for b, and -15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}-4\left(-2\right)\left(-15\right)}}{2\left(-2\right)}$$

Square 13.

$$x=\frac{-13±\sqrt{169-4\left(-2\right)\left(-15\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-13±\sqrt{169+8\left(-15\right)}}{2\left(-2\right)}$$

Multiply 8 times -15.

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

Add 169 to -120.

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

Take the square root of 49.

$$x=\frac{-13±7}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-13±7}{-4}$$

Now solve the equation x=\frac{-13±7}{-4} when ± is plus. Add -13 to 7.

$$x=-\frac{6}{-4}$$

Reduce the fraction \frac{-6}{-4} to lowest terms by extracting and canceling out 2.

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

Now solve the equation x=\frac{-13±7}{-4} when ± is minus. Subtract 7 from -13.

$$x=-\frac{20}{-4}$$

Divide -20 by -4.

$$x=5$$

The equation is now solved.

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

Steps for Completing the Square

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

$$\left(x-1\right)\left(x+1\right)=\left(x-2\right)\left(3x-7\right)$$

Consider \left(x-1\right)\left(x+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

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

Use the distributive property to multiply x-2 by 3x-7 and combine like terms.

$$x^{2}-1=3x^{2}-13x+14$$

Subtract 3x^{2} from both sides.

$$x^{2}-1-3x^{2}=-13x+14$$

Combine x^{2} and -3x^{2} to get -2x^{2}.

$$-2x^{2}-1=-13x+14$$

Add 13x to both sides.

$$-2x^{2}-1+13x=14$$

Add 1 to both sides.

$$-2x^{2}+13x=14+1$$

Add 14 and 1 to get 15.

$$-2x^{2}+13x=15$$

Divide both sides by -2.

$$\frac{-2x^{2}+13x}{-2}=\frac{15}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{13}{-2}x=\frac{15}{-2}$$

Divide 13 by -2.

$$x^{2}-\frac{13}{2}x=\frac{15}{-2}$$

Divide 15 by -2.

$$x^{2}-\frac{13}{2}x=-\frac{15}{2}$$

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.

$$x^{2}-\frac{13}{2}x+\left(-\frac{13}{4}\right)^{2}=-\frac{15}{2}+\left(-\frac{13}{4}\right)^{2}$$

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

$$x^{2}-\frac{13}{2}x+\frac{169}{16}=-\frac{15}{2}+\frac{169}{16}$$

Add -\frac{15}{2} to \frac{169}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{13}{2}x+\frac{169}{16}=\frac{49}{16}$$

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}.

$$\left(x-\frac{13}{4}\right)^{2}=\frac{49}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{4}\right)^{2}}=\sqrt{\frac{49}{16}}$$

Simplify.

$$x-\frac{13}{4}=\frac{7}{4} x-\frac{13}{4}=-\frac{7}{4}$$

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

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