Solve 3x-5/2x-5=x-7/x-4

$$\frac { 3 x - 5 } { 2 x - 5 } = \frac { x - 7 } { x - 4 }$$

Solve for x

$$x=-5$$

$$x=3$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values \frac{5}{2},4 since division by zero is not defined. Multiply both sides of the equation by \left(x-4\right)\left(2x-5\right), the least common multiple of 2x-5,x-4.

$$\left(x-4\right)\left(3x-5\right)=\left(2x-5\right)\left(x-7\right)$$

Use the distributive property to multiply x-4 by 3x-5 and combine like terms.

$$3x^{2}-17x+20=\left(2x-5\right)\left(x-7\right)$$

Use the distributive property to multiply 2x-5 by x-7 and combine like terms.

$$3x^{2}-17x+20=2x^{2}-19x+35$$

Subtract 2x^{2} from both sides.

$$3x^{2}-17x+20-2x^{2}=-19x+35$$

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

$$x^{2}-17x+20=-19x+35$$

Add 19x to both sides.

$$x^{2}-17x+20+19x=35$$

Combine -17x and 19x to get 2x.

$$x^{2}+2x+20=35$$

Subtract 35 from both sides.

$$x^{2}+2x+20-35=0$$

Subtract 35 from 20 to get -15.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and -15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square 2.

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

Multiply -4 times -15.

$$x=\frac{-2±\sqrt{4+60}}{2}$$

Add 4 to 60.

$$x=\frac{-2±\sqrt{64}}{2}$$

Take the square root of 64.

$$x=\frac{-2±8}{2}$$

Now solve the equation x=\frac{-2±8}{2} when ± is plus. Add -2 to 8.

$$x=\frac{6}{2}$$

Divide 6 by 2.

$$x=3$$

Now solve the equation x=\frac{-2±8}{2} when ± is minus. Subtract 8 from -2.

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

Divide -10 by 2.

$$x=-5$$

The equation is now solved.

$$x=3 x=-5$$

Steps for Completing the Square

Variable x cannot be equal to any of the values \frac{5}{2},4 since division by zero is not defined. Multiply both sides of the equation by \left(x-4\right)\left(2x-5\right), the least common multiple of 2x-5,x-4.

$$\left(x-4\right)\left(3x-5\right)=\left(2x-5\right)\left(x-7\right)$$

Use the distributive property to multiply x-4 by 3x-5 and combine like terms.

$$3x^{2}-17x+20=\left(2x-5\right)\left(x-7\right)$$

Use the distributive property to multiply 2x-5 by x-7 and combine like terms.

$$3x^{2}-17x+20=2x^{2}-19x+35$$

Subtract 2x^{2} from both sides.

$$3x^{2}-17x+20-2x^{2}=-19x+35$$

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

$$x^{2}-17x+20=-19x+35$$

Add 19x to both sides.

$$x^{2}-17x+20+19x=35$$

Combine -17x and 19x to get 2x.

$$x^{2}+2x+20=35$$

Subtract 20 from both sides.

$$x^{2}+2x=35-20$$

Subtract 20 from 35 to get 15.

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

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

$$x^{2}+2x+1^{2}=15+1^{2}$$

Square 1.

$$x^{2}+2x+1=15+1$$

Add 15 to 1.

$$x^{2}+2x+1=16$$

Factor x^{2}+2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1\right)^{2}=16$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{16}$$

Simplify.

$$x+1=4 x+1=-4$$

Subtract 1 from both sides of the equation.

$$x=3 x=-5$$