Solve 2x-3/4-x/3=1/x+3-25/12

$$\frac { 2 x - 3 } { 4 } - \frac { x } { 3 } = \frac { 1 } { x + 3 } - \frac { 25 } { 12 }$$

Solve for x

$$x=-9$$

$$x=-2$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -3 since division by zero is not defined. Multiply both sides of the equation by 12\left(x+3\right), the least common multiple of 4,3,x+3,12.

$$\left(3x+9\right)\left(2x-3\right)-\left(4x+12\right)x=12+12\left(x+3\right)\left(-\frac{25}{12}\right)$$

Use the distributive property to multiply 3x+9 by 2x-3 and combine like terms.

$$6x^{2}+9x-27-\left(4x+12\right)x=12+12\left(x+3\right)\left(-\frac{25}{12}\right)$$

Use the distributive property to multiply 4x+12 by x.

$$6x^{2}+9x-27-\left(4x^{2}+12x\right)=12+12\left(x+3\right)\left(-\frac{25}{12}\right)$$

To find the opposite of 4x^{2}+12x, find the opposite of each term.

$$6x^{2}+9x-27-4x^{2}-12x=12+12\left(x+3\right)\left(-\frac{25}{12}\right)$$

Combine 6x^{2} and -4x^{2} to get 2x^{2}.

$$2x^{2}+9x-27-12x=12+12\left(x+3\right)\left(-\frac{25}{12}\right)$$

Combine 9x and -12x to get -3x.

$$2x^{2}-3x-27=12+12\left(x+3\right)\left(-\frac{25}{12}\right)$$

Multiply 12 and -\frac{25}{12} to get -25.

$$2x^{2}-3x-27=12-25\left(x+3\right)$$

Use the distributive property to multiply -25 by x+3.

$$2x^{2}-3x-27=12-25x-75$$

Subtract 75 from 12 to get -63.

$$2x^{2}-3x-27=-63-25x$$

Subtract -63 from both sides.

$$2x^{2}-3x-27-\left(-63\right)=-25x$$

The opposite of -63 is 63.

$$2x^{2}-3x-27+63=-25x$$

Add 25x to both sides.

$$2x^{2}-3x-27+63+25x=0$$

Add -27 and 63 to get 36.

$$2x^{2}-3x+36+25x=0$$

Combine -3x and 25x to get 22x.

$$2x^{2}+22x+36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 22 for b, and 36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-22±\sqrt{22^{2}-4\times 2\times 36}}{2\times 2}$$

Square 22.

$$x=\frac{-22±\sqrt{484-4\times 2\times 36}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-22±\sqrt{484-8\times 36}}{2\times 2}$$

Multiply -8 times 36.

$$x=\frac{-22±\sqrt{484-288}}{2\times 2}$$

Add 484 to -288.

$$x=\frac{-22±\sqrt{196}}{2\times 2}$$

Take the square root of 196.

$$x=\frac{-22±14}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-22±14}{4}$$

Now solve the equation x=\frac{-22±14}{4} when ± is plus. Add -22 to 14.

$$x=-\frac{8}{4}$$

Divide -8 by 4.

$$x=-2$$

Now solve the equation x=\frac{-22±14}{4} when ± is minus. Subtract 14 from -22.

$$x=-\frac{36}{4}$$

Divide -36 by 4.

$$x=-9$$

The equation is now solved.

$$x=-2 x=-9$$

Steps for Completing the Square

Variable x cannot be equal to -3 since division by zero is not defined. Multiply both sides of the equation by 12\left(x+3\right), the least common multiple of 4,3,x+3,12.

$$\left(3x+9\right)\left(2x-3\right)-\left(4x+12\right)x=12+12\left(x+3\right)\left(-\frac{25}{12}\right)$$

Use the distributive property to multiply 3x+9 by 2x-3 and combine like terms.

$$6x^{2}+9x-27-\left(4x+12\right)x=12+12\left(x+3\right)\left(-\frac{25}{12}\right)$$

Use the distributive property to multiply 4x+12 by x.

$$6x^{2}+9x-27-\left(4x^{2}+12x\right)=12+12\left(x+3\right)\left(-\frac{25}{12}\right)$$

To find the opposite of 4x^{2}+12x, find the opposite of each term.

$$6x^{2}+9x-27-4x^{2}-12x=12+12\left(x+3\right)\left(-\frac{25}{12}\right)$$

Combine 6x^{2} and -4x^{2} to get 2x^{2}.

$$2x^{2}+9x-27-12x=12+12\left(x+3\right)\left(-\frac{25}{12}\right)$$

Combine 9x and -12x to get -3x.

$$2x^{2}-3x-27=12+12\left(x+3\right)\left(-\frac{25}{12}\right)$$

Multiply 12 and -\frac{25}{12} to get -25.

$$2x^{2}-3x-27=12-25\left(x+3\right)$$

Use the distributive property to multiply -25 by x+3.

$$2x^{2}-3x-27=12-25x-75$$

Subtract 75 from 12 to get -63.

$$2x^{2}-3x-27=-63-25x$$

Add 25x to both sides.

$$2x^{2}-3x-27+25x=-63$$

Combine -3x and 25x to get 22x.

$$2x^{2}+22x-27=-63$$

Add 27 to both sides.

$$2x^{2}+22x=-63+27$$

Add -63 and 27 to get -36.

$$2x^{2}+22x=-36$$

Divide both sides by 2.

$$\frac{2x^{2}+22x}{2}=-\frac{36}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{22}{2}x=-\frac{36}{2}$$

Divide 22 by 2.

$$x^{2}+11x=-\frac{36}{2}$$

Divide -36 by 2.

$$x^{2}+11x=-18$$

Divide 11, the coefficient of the x term, by 2 to get \frac{11}{2}. Then add the square of \frac{11}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+11x+\left(\frac{11}{2}\right)^{2}=-18+\left(\frac{11}{2}\right)^{2}$$

Square \frac{11}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+11x+\frac{121}{4}=-18+\frac{121}{4}$$

Add -18 to \frac{121}{4}.

$$x^{2}+11x+\frac{121}{4}=\frac{49}{4}$$

Factor x^{2}+11x+\frac{121}{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}.

$$\left(x+\frac{11}{2}\right)^{2}=\frac{49}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{11}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Simplify.

$$x+\frac{11}{2}=\frac{7}{2} x+\frac{11}{2}=-\frac{7}{2}$$

Subtract \frac{11}{2} from both sides of the equation.

$$x=-2 x=-9$$