Solve x-1/x+3-x-4/2(x-3)=1/4

$$\frac { x - 1 } { x + 3 } - \frac { x - 4 } { 2 ( x - 3 ) } = \frac { 1 } { 4 }$$

Solve for x

$$x=5$$

$$x=9$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -3,3 since division by zero is not defined. Multiply both sides of the equation by 4\left(x-3\right)\left(x+3\right), the least common multiple of x+3,2\left(x-3\right),4.

$$\left(4x-12\right)\left(x-1\right)-\left(2x+6\right)\left(x-4\right)=\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply 4x-12 by x-1 and combine like terms.

$$4x^{2}-16x+12-\left(2x+6\right)\left(x-4\right)=\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply 2x+6 by x-4 and combine like terms.

$$4x^{2}-16x+12-\left(2x^{2}-2x-24\right)=\left(x-3\right)\left(x+3\right)$$

To find the opposite of 2x^{2}-2x-24, find the opposite of each term.

$$4x^{2}-16x+12-2x^{2}+2x+24=\left(x-3\right)\left(x+3\right)$$

Combine 4x^{2} and -2x^{2} to get 2x^{2}.

$$2x^{2}-16x+12+2x+24=\left(x-3\right)\left(x+3\right)$$

Combine -16x and 2x to get -14x.

$$2x^{2}-14x+12+24=\left(x-3\right)\left(x+3\right)$$

Add 12 and 24 to get 36.

$$2x^{2}-14x+36=\left(x-3\right)\left(x+3\right)$$

Consider \left(x-3\right)\left(x+3\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 3.

$$2x^{2}-14x+36=x^{2}-9$$

Subtract x^{2} from both sides.

$$2x^{2}-14x+36-x^{2}=-9$$

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

$$x^{2}-14x+36=-9$$

Add 9 to both sides.

$$x^{2}-14x+36+9=0$$

Add 36 and 9 to get 45.

$$x^{2}-14x+45=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -14 for b, and 45 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 45}}{2}$$

Square -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 45}}{2}$$

Multiply -4 times 45.

$$x=\frac{-\left(-14\right)±\sqrt{196-180}}{2}$$

Add 196 to -180.

$$x=\frac{-\left(-14\right)±\sqrt{16}}{2}$$

Take the square root of 16.

$$x=\frac{-\left(-14\right)±4}{2}$$

The opposite of -14 is 14.

$$x=\frac{14±4}{2}$$

Now solve the equation x=\frac{14±4}{2} when ± is plus. Add 14 to 4.

$$x=\frac{18}{2}$$

Divide 18 by 2.

$$x=9$$

Now solve the equation x=\frac{14±4}{2} when ± is minus. Subtract 4 from 14.

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

Divide 10 by 2.

$$x=5$$

The equation is now solved.

$$x=9 x=5$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -3,3 since division by zero is not defined. Multiply both sides of the equation by 4\left(x-3\right)\left(x+3\right), the least common multiple of x+3,2\left(x-3\right),4.

$$\left(4x-12\right)\left(x-1\right)-\left(2x+6\right)\left(x-4\right)=\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply 4x-12 by x-1 and combine like terms.

$$4x^{2}-16x+12-\left(2x+6\right)\left(x-4\right)=\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply 2x+6 by x-4 and combine like terms.

$$4x^{2}-16x+12-\left(2x^{2}-2x-24\right)=\left(x-3\right)\left(x+3\right)$$

To find the opposite of 2x^{2}-2x-24, find the opposite of each term.

$$4x^{2}-16x+12-2x^{2}+2x+24=\left(x-3\right)\left(x+3\right)$$

Combine 4x^{2} and -2x^{2} to get 2x^{2}.

$$2x^{2}-16x+12+2x+24=\left(x-3\right)\left(x+3\right)$$

Combine -16x and 2x to get -14x.

$$2x^{2}-14x+12+24=\left(x-3\right)\left(x+3\right)$$

Add 12 and 24 to get 36.

$$2x^{2}-14x+36=\left(x-3\right)\left(x+3\right)$$

Consider \left(x-3\right)\left(x+3\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 3.

$$2x^{2}-14x+36=x^{2}-9$$

Subtract x^{2} from both sides.

$$2x^{2}-14x+36-x^{2}=-9$$

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

$$x^{2}-14x+36=-9$$

Subtract 36 from both sides.

$$x^{2}-14x=-9-36$$

Subtract 36 from -9 to get -45.

$$x^{2}-14x=-45$$

Divide -14, the coefficient of the x term, by 2 to get -7. Then add the square of -7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-14x+\left(-7\right)^{2}=-45+\left(-7\right)^{2}$$

Square -7.

$$x^{2}-14x+49=-45+49$$

Add -45 to 49.

$$x^{2}-14x+49=4$$

Factor x^{2}-14x+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-7\right)^{2}=4$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{4}$$

Simplify.

$$x-7=2 x-7=-2$$

Add 7 to both sides of the equation.

$$x=9 x=5$$