Solve 4/x+3-5/3-x=1/x-3-1

$$\frac { 4 } { x + 3 } - \frac { 5 } { 3 - x } = \frac { 1 } { x - 3 } - 1$$

Solve for x

$$x=-9$$

$$x=1$$

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 \left(x-3\right)\left(x+3\right), the least common multiple of x+3,3-x,x-3.

$$\left(x-3\right)\times 4-\left(-\left(3+x\right)\times 5\right)=x+3+\left(x-3\right)\left(x+3\right)\left(-1\right)$$

Use the distributive property to multiply x-3 by 4.

$$4x-12-\left(-\left(3+x\right)\times 5\right)=x+3+\left(x-3\right)\left(x+3\right)\left(-1\right)$$

Multiply -1 and 5 to get -5.

$$4x-12-\left(-5\left(3+x\right)\right)=x+3+\left(x-3\right)\left(x+3\right)\left(-1\right)$$

Use the distributive property to multiply -5 by 3+x.

$$4x-12-\left(-15-5x\right)=x+3+\left(x-3\right)\left(x+3\right)\left(-1\right)$$

To find the opposite of -15-5x, find the opposite of each term.

$$4x-12+15+5x=x+3+\left(x-3\right)\left(x+3\right)\left(-1\right)$$

Add -12 and 15 to get 3.

$$4x+3+5x=x+3+\left(x-3\right)\left(x+3\right)\left(-1\right)$$

Combine 4x and 5x to get 9x.

$$9x+3=x+3+\left(x-3\right)\left(x+3\right)\left(-1\right)$$

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

$$9x+3=x+3+\left(x^{2}-9\right)\left(-1\right)$$

Use the distributive property to multiply x^{2}-9 by -1.

$$9x+3=x+3-x^{2}+9$$

Add 3 and 9 to get 12.

$$9x+3=x+12-x^{2}$$

Subtract x from both sides.

$$9x+3-x=12-x^{2}$$

Combine 9x and -x to get 8x.

$$8x+3=12-x^{2}$$

Subtract 12 from both sides.

$$8x+3-12=-x^{2}$$

Subtract 12 from 3 to get -9.

$$8x-9=-x^{2}$$

Add x^{2} to both sides.

$$8x-9+x^{2}=0$$

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.

$$x^{2}+8x-9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 8 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-9\right)}}{2}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\left(-9\right)}}{2}$$

Multiply -4 times -9.

$$x=\frac{-8±\sqrt{64+36}}{2}$$

Add 64 to 36.

$$x=\frac{-8±\sqrt{100}}{2}$$

Take the square root of 100.

$$x=\frac{-8±10}{2}$$

Now solve the equation x=\frac{-8±10}{2} when ± is plus. Add -8 to 10.

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

Divide 2 by 2.

$$x=1$$

Now solve the equation x=\frac{-8±10}{2} when ± is minus. Subtract 10 from -8.

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

Divide -18 by 2.

$$x=-9$$

The equation is now solved.

$$x=1 x=-9$$

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 \left(x-3\right)\left(x+3\right), the least common multiple of x+3,3-x,x-3.

$$\left(x-3\right)\times 4-\left(-\left(3+x\right)\times 5\right)=x+3+\left(x-3\right)\left(x+3\right)\left(-1\right)$$

Use the distributive property to multiply x-3 by 4.

$$4x-12-\left(-\left(3+x\right)\times 5\right)=x+3+\left(x-3\right)\left(x+3\right)\left(-1\right)$$

Multiply -1 and 5 to get -5.

$$4x-12-\left(-5\left(3+x\right)\right)=x+3+\left(x-3\right)\left(x+3\right)\left(-1\right)$$

Use the distributive property to multiply -5 by 3+x.

$$4x-12-\left(-15-5x\right)=x+3+\left(x-3\right)\left(x+3\right)\left(-1\right)$$

To find the opposite of -15-5x, find the opposite of each term.

$$4x-12+15+5x=x+3+\left(x-3\right)\left(x+3\right)\left(-1\right)$$

Add -12 and 15 to get 3.

$$4x+3+5x=x+3+\left(x-3\right)\left(x+3\right)\left(-1\right)$$

Combine 4x and 5x to get 9x.

$$9x+3=x+3+\left(x-3\right)\left(x+3\right)\left(-1\right)$$

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

$$9x+3=x+3+\left(x^{2}-9\right)\left(-1\right)$$

Use the distributive property to multiply x^{2}-9 by -1.

$$9x+3=x+3-x^{2}+9$$

Add 3 and 9 to get 12.

$$9x+3=x+12-x^{2}$$

Subtract x from both sides.

$$9x+3-x=12-x^{2}$$

Combine 9x and -x to get 8x.

$$8x+3=12-x^{2}$$

Add x^{2} to both sides.

$$8x+3+x^{2}=12$$

Subtract 3 from both sides.

$$8x+x^{2}=12-3$$

Subtract 3 from 12 to get 9.

$$8x+x^{2}=9$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}+8x=9$$

Divide 8, the coefficient of the x term, by 2 to get 4. Then add the square of 4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+8x+4^{2}=9+4^{2}$$

Square 4.

$$x^{2}+8x+16=9+16$$

Add 9 to 16.

$$x^{2}+8x+16=25$$

Factor x^{2}+8x+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}.

$$\left(x+4\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{25}$$

Simplify.

$$x+4=5 x+4=-5$$

Subtract 4 from both sides of the equation.

$$x=1 x=-9$$