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

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

Solve for x

$$x=-1$$

$$x=12$$

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,x+3.

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

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

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

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

$$4x^{2}+12x+\left(x-3\right)\left(x+3\right)\left(-3\right)=\left(x-3\right)\left(3x-1\right)$$

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

$$4x^{2}+12x+\left(x^{2}-9\right)\left(-3\right)=\left(x-3\right)\left(3x-1\right)$$

Use the distributive property to multiply x^{2}-9 by -3.

$$4x^{2}+12x-3x^{2}+27=\left(x-3\right)\left(3x-1\right)$$

Combine 4x^{2} and -3x^{2} to get x^{2}.

$$x^{2}+12x+27=\left(x-3\right)\left(3x-1\right)$$

Use the distributive property to multiply x-3 by 3x-1 and combine like terms.

$$x^{2}+12x+27=3x^{2}-10x+3$$

Subtract 3x^{2} from both sides.

$$x^{2}+12x+27-3x^{2}=-10x+3$$

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

$$-2x^{2}+12x+27=-10x+3$$

Add 10x to both sides.

$$-2x^{2}+12x+27+10x=3$$

Combine 12x and 10x to get 22x.

$$-2x^{2}+22x+27=3$$

Subtract 3 from both sides.

$$-2x^{2}+22x+27-3=0$$

Subtract 3 from 27 to get 24.

$$-2x^{2}+22x+24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 22 for b, and 24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-22±\sqrt{22^{2}-4\left(-2\right)\times 24}}{2\left(-2\right)}$$

Square 22.

$$x=\frac{-22±\sqrt{484-4\left(-2\right)\times 24}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-22±\sqrt{484+8\times 24}}{2\left(-2\right)}$$

Multiply 8 times 24.

$$x=\frac{-22±\sqrt{484+192}}{2\left(-2\right)}$$

Add 484 to 192.

$$x=\frac{-22±\sqrt{676}}{2\left(-2\right)}$$

Take the square root of 676.

$$x=\frac{-22±26}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-22±26}{-4}$$

Now solve the equation x=\frac{-22±26}{-4} when ± is plus. Add -22 to 26.

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

Divide 4 by -4.

$$x=-1$$

Now solve the equation x=\frac{-22±26}{-4} when ± is minus. Subtract 26 from -22.

$$x=-\frac{48}{-4}$$

Divide -48 by -4.

$$x=12$$

The equation is now solved.

$$x=-1 x=12$$

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,x+3.

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

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

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

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

$$4x^{2}+12x+\left(x-3\right)\left(x+3\right)\left(-3\right)=\left(x-3\right)\left(3x-1\right)$$

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

$$4x^{2}+12x+\left(x^{2}-9\right)\left(-3\right)=\left(x-3\right)\left(3x-1\right)$$

Use the distributive property to multiply x^{2}-9 by -3.

$$4x^{2}+12x-3x^{2}+27=\left(x-3\right)\left(3x-1\right)$$

Combine 4x^{2} and -3x^{2} to get x^{2}.

$$x^{2}+12x+27=\left(x-3\right)\left(3x-1\right)$$

Use the distributive property to multiply x-3 by 3x-1 and combine like terms.

$$x^{2}+12x+27=3x^{2}-10x+3$$

Subtract 3x^{2} from both sides.

$$x^{2}+12x+27-3x^{2}=-10x+3$$

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

$$-2x^{2}+12x+27=-10x+3$$

Add 10x to both sides.

$$-2x^{2}+12x+27+10x=3$$

Combine 12x and 10x to get 22x.

$$-2x^{2}+22x+27=3$$

Subtract 27 from both sides.

$$-2x^{2}+22x=3-27$$

Subtract 27 from 3 to get -24.

$$-2x^{2}+22x=-24$$

Divide both sides by -2.

$$\frac{-2x^{2}+22x}{-2}=-\frac{24}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{22}{-2}x=-\frac{24}{-2}$$

Divide 22 by -2.

$$x^{2}-11x=-\frac{24}{-2}$$

Divide -24 by -2.

$$x^{2}-11x=12$$

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.

$$x^{2}-11x+\left(-\frac{11}{2}\right)^{2}=12+\left(-\frac{11}{2}\right)^{2}$$

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

$$x^{2}-11x+\frac{121}{4}=12+\frac{121}{4}$$

Add 12 to \frac{121}{4}.

$$x^{2}-11x+\frac{121}{4}=\frac{169}{4}$$

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}.

$$\left(x-\frac{11}{2}\right)^{2}=\frac{169}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{11}{2}\right)^{2}}=\sqrt{\frac{169}{4}}$$

Simplify.

$$x-\frac{11}{2}=\frac{13}{2} x-\frac{11}{2}=-\frac{13}{2}$$

Add \frac{11}{2} to both sides of the equation.

$$x=12 x=-1$$