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

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

Solve for x

$$x=3$$

$$x=12$$

Steps Using the Quadratic Formula

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

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

Use the distributive property to multiply x+4 by 2x-7 and combine like terms.

$$2x^{2}+x-28=\left(x-4\right)\left(3x-2\right)$$

Use the distributive property to multiply x-4 by 3x-2 and combine like terms.

$$2x^{2}+x-28=3x^{2}-14x+8$$

Subtract 3x^{2} from both sides.

$$2x^{2}+x-28-3x^{2}=-14x+8$$

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

$$-x^{2}+x-28=-14x+8$$

Add 14x to both sides.

$$-x^{2}+x-28+14x=8$$

Combine x and 14x to get 15x.

$$-x^{2}+15x-28=8$$

Subtract 8 from both sides.

$$-x^{2}+15x-28-8=0$$

Subtract 8 from -28 to get -36.

$$-x^{2}+15x-36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 15 for b, and -36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-15±\sqrt{15^{2}-4\left(-1\right)\left(-36\right)}}{2\left(-1\right)}$$

Square 15.

$$x=\frac{-15±\sqrt{225-4\left(-1\right)\left(-36\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-15±\sqrt{225+4\left(-36\right)}}{2\left(-1\right)}$$

Multiply 4 times -36.

$$x=\frac{-15±\sqrt{225-144}}{2\left(-1\right)}$$

Add 225 to -144.

$$x=\frac{-15±\sqrt{81}}{2\left(-1\right)}$$

Take the square root of 81.

$$x=\frac{-15±9}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-15±9}{-2}$$

Now solve the equation x=\frac{-15±9}{-2} when ± is plus. Add -15 to 9.

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

Divide -6 by -2.

$$x=3$$

Now solve the equation x=\frac{-15±9}{-2} when ± is minus. Subtract 9 from -15.

$$x=-\frac{24}{-2}$$

Divide -24 by -2.

$$x=12$$

The equation is now solved.

$$x=3 x=12$$

Steps for Completing the Square

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

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

Use the distributive property to multiply x+4 by 2x-7 and combine like terms.

$$2x^{2}+x-28=\left(x-4\right)\left(3x-2\right)$$

Use the distributive property to multiply x-4 by 3x-2 and combine like terms.

$$2x^{2}+x-28=3x^{2}-14x+8$$

Subtract 3x^{2} from both sides.

$$2x^{2}+x-28-3x^{2}=-14x+8$$

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

$$-x^{2}+x-28=-14x+8$$

Add 14x to both sides.

$$-x^{2}+x-28+14x=8$$

Combine x and 14x to get 15x.

$$-x^{2}+15x-28=8$$

Add 28 to both sides.

$$-x^{2}+15x=8+28$$

Add 8 and 28 to get 36.

$$-x^{2}+15x=36$$

Divide both sides by -1.

$$\frac{-x^{2}+15x}{-1}=\frac{36}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{15}{-1}x=\frac{36}{-1}$$

Divide 15 by -1.

$$x^{2}-15x=\frac{36}{-1}$$

Divide 36 by -1.

$$x^{2}-15x=-36$$

Divide -15, the coefficient of the x term, by 2 to get -\frac{15}{2}. Then add the square of -\frac{15}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-15x+\left(-\frac{15}{2}\right)^{2}=-36+\left(-\frac{15}{2}\right)^{2}$$

Square -\frac{15}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-15x+\frac{225}{4}=-36+\frac{225}{4}$$

Add -36 to \frac{225}{4}.

$$x^{2}-15x+\frac{225}{4}=\frac{81}{4}$$

Factor x^{2}-15x+\frac{225}{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}.

$$\left(x-\frac{15}{2}\right)^{2}=\frac{81}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{15}{2}\right)^{2}}=\sqrt{\frac{81}{4}}$$

Simplify.

$$x-\frac{15}{2}=\frac{9}{2} x-\frac{15}{2}=-\frac{9}{2}$$

Add \frac{15}{2} to both sides of the equation.

$$x=12 x=3$$