Solve -2/w-3=7/4w-6+4

$$- \frac { 2 } { w - 3 } = \frac { 7 } { 4 w - 6 } + 4$$

Solve for w

$$w = \frac{\sqrt{753} + 57}{32} \approx 2.638776421$$

$$w=\frac{57-\sqrt{753}}{32}\approx 0.923723579$$

Steps Using the Quadratic Formula

Variable w cannot be equal to any of the values \frac{3}{2},3 since division by zero is not defined. Multiply both sides of the equation by 2\left(w-3\right)\left(2w-3\right), the least common multiple of w-3,4w-6.

$$-\left(4w-6\right)\times 2=\left(w-3\right)\times 7+2\left(w-3\right)\left(2w-3\right)\times 4$$

Use the distributive property to multiply 4w-6 by 2.

$$-\left(8w-12\right)=\left(w-3\right)\times 7+2\left(w-3\right)\left(2w-3\right)\times 4$$

To find the opposite of 8w-12, find the opposite of each term.

$$-8w+12=\left(w-3\right)\times 7+2\left(w-3\right)\left(2w-3\right)\times 4$$

Use the distributive property to multiply w-3 by 7.

$$-8w+12=7w-21+2\left(w-3\right)\left(2w-3\right)\times 4$$

Multiply 2 and 4 to get 8.

$$-8w+12=7w-21+8\left(w-3\right)\left(2w-3\right)$$

Use the distributive property to multiply 8 by w-3.

$$-8w+12=7w-21+\left(8w-24\right)\left(2w-3\right)$$

Use the distributive property to multiply 8w-24 by 2w-3 and combine like terms.

$$-8w+12=7w-21+16w^{2}-72w+72$$

Combine 7w and -72w to get -65w.

$$-8w+12=-65w-21+16w^{2}+72$$

Add -21 and 72 to get 51.

$$-8w+12=-65w+51+16w^{2}$$

Add 65w to both sides.

$$-8w+12+65w=51+16w^{2}$$

Combine -8w and 65w to get 57w.

$$57w+12=51+16w^{2}$$

Subtract 51 from both sides.

$$57w+12-51=16w^{2}$$

Subtract 51 from 12 to get -39.

$$57w-39=16w^{2}$$

Subtract 16w^{2} from both sides.

$$57w-39-16w^{2}=0$$

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.

$$-16w^{2}+57w-39=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 57 for b, and -39 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$w=\frac{-57±\sqrt{57^{2}-4\left(-16\right)\left(-39\right)}}{2\left(-16\right)}$$

Square 57.

$$w=\frac{-57±\sqrt{3249-4\left(-16\right)\left(-39\right)}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$w=\frac{-57±\sqrt{3249+64\left(-39\right)}}{2\left(-16\right)}$$

Multiply 64 times -39.

$$w=\frac{-57±\sqrt{3249-2496}}{2\left(-16\right)}$$

Add 3249 to -2496.

$$w=\frac{-57±\sqrt{753}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$w=\frac{-57±\sqrt{753}}{-32}$$

Now solve the equation w=\frac{-57±\sqrt{753}}{-32} when ± is plus. Add -57 to \sqrt{753}.

$$w=\frac{\sqrt{753}-57}{-32}$$

Divide -57+\sqrt{753} by -32.

$$w=\frac{57-\sqrt{753}}{32}$$

Now solve the equation w=\frac{-57±\sqrt{753}}{-32} when ± is minus. Subtract \sqrt{753} from -57.

$$w=\frac{-\sqrt{753}-57}{-32}$$

Divide -57-\sqrt{753} by -32.

$$w=\frac{\sqrt{753}+57}{32}$$

The equation is now solved.

$$w=\frac{57-\sqrt{753}}{32} w=\frac{\sqrt{753}+57}{32}$$

Steps for Completing the Square

Variable w cannot be equal to any of the values \frac{3}{2},3 since division by zero is not defined. Multiply both sides of the equation by 2\left(w-3\right)\left(2w-3\right), the least common multiple of w-3,4w-6.

$$-\left(4w-6\right)\times 2=\left(w-3\right)\times 7+2\left(w-3\right)\left(2w-3\right)\times 4$$

Use the distributive property to multiply 4w-6 by 2.

$$-\left(8w-12\right)=\left(w-3\right)\times 7+2\left(w-3\right)\left(2w-3\right)\times 4$$

To find the opposite of 8w-12, find the opposite of each term.

$$-8w+12=\left(w-3\right)\times 7+2\left(w-3\right)\left(2w-3\right)\times 4$$

Use the distributive property to multiply w-3 by 7.

$$-8w+12=7w-21+2\left(w-3\right)\left(2w-3\right)\times 4$$

Multiply 2 and 4 to get 8.

$$-8w+12=7w-21+8\left(w-3\right)\left(2w-3\right)$$

Use the distributive property to multiply 8 by w-3.

$$-8w+12=7w-21+\left(8w-24\right)\left(2w-3\right)$$

Use the distributive property to multiply 8w-24 by 2w-3 and combine like terms.

$$-8w+12=7w-21+16w^{2}-72w+72$$

Combine 7w and -72w to get -65w.

$$-8w+12=-65w-21+16w^{2}+72$$

Add -21 and 72 to get 51.

$$-8w+12=-65w+51+16w^{2}$$

Add 65w to both sides.

$$-8w+12+65w=51+16w^{2}$$

Combine -8w and 65w to get 57w.

$$57w+12=51+16w^{2}$$

Subtract 16w^{2} from both sides.

$$57w+12-16w^{2}=51$$

Subtract 12 from both sides.

$$57w-16w^{2}=51-12$$

Subtract 12 from 51 to get 39.

$$57w-16w^{2}=39$$

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.

$$-16w^{2}+57w=39$$

Divide both sides by -16.

$$\frac{-16w^{2}+57w}{-16}=\frac{39}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$w^{2}+\frac{57}{-16}w=\frac{39}{-16}$$

Divide 57 by -16.

$$w^{2}-\frac{57}{16}w=\frac{39}{-16}$$

Divide 39 by -16.

$$w^{2}-\frac{57}{16}w=-\frac{39}{16}$$

Divide -\frac{57}{16}, the coefficient of the x term, by 2 to get -\frac{57}{32}. Then add the square of -\frac{57}{32} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$w^{2}-\frac{57}{16}w+\left(-\frac{57}{32}\right)^{2}=-\frac{39}{16}+\left(-\frac{57}{32}\right)^{2}$$

Square -\frac{57}{32} by squaring both the numerator and the denominator of the fraction.

$$w^{2}-\frac{57}{16}w+\frac{3249}{1024}=-\frac{39}{16}+\frac{3249}{1024}$$

Add -\frac{39}{16} to \frac{3249}{1024} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$w^{2}-\frac{57}{16}w+\frac{3249}{1024}=\frac{753}{1024}$$

Factor w^{2}-\frac{57}{16}w+\frac{3249}{1024}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(w-\frac{57}{32}\right)^{2}=\frac{753}{1024}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(w-\frac{57}{32}\right)^{2}}=\sqrt{\frac{753}{1024}}$$

Simplify.

$$w-\frac{57}{32}=\frac{\sqrt{753}}{32} w-\frac{57}{32}=-\frac{\sqrt{753}}{32}$$

Add \frac{57}{32} to both sides of the equation.

$$w=\frac{\sqrt{753}+57}{32} w=\frac{57-\sqrt{753}}{32}$$