Solve 3/w^2+5w+6+w-1/w+2=7/w+3

$$\frac { 3 } { w ^ { 2 } + 5 w + 6 } + \frac { w - 1 } { w + 2 } = \frac { 7 } { w + 3 }$$

Solve for w

$$w=7$$

Steps Using Factoring

Variable w cannot be equal to any of the values -3,-2 since division by zero is not defined. Multiply both sides of the equation by \left(w+2\right)\left(w+3\right), the least common multiple of w^{2}+5w+6,w+2,w+3.

$$3+\left(w+3\right)\left(w-1\right)=\left(w+2\right)\times 7$$

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

$$3+w^{2}+2w-3=\left(w+2\right)\times 7$$

Subtract 3 from 3 to get 0.

$$w^{2}+2w=\left(w+2\right)\times 7$$

Use the distributive property to multiply w+2 by 7.

$$w^{2}+2w=7w+14$$

Subtract 7w from both sides.

$$w^{2}+2w-7w=14$$

Combine 2w and -7w to get -5w.

$$w^{2}-5w=14$$

Subtract 14 from both sides.

$$w^{2}-5w-14=0$$

To solve the equation, factor w^{2}-5w-14 using formula w^{2}+\left(a+b\right)w+ab=\left(w+a\right)\left(w+b\right). To find a and b, set up a system to be solved.

$$a+b=-5 ab=-14$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -14.

$$1,-14 2,-7$$

Calculate the sum for each pair.

$$1-14=-13 2-7=-5$$

The solution is the pair that gives sum -5.

$$a=-7 b=2$$

Rewrite factored expression \left(w+a\right)\left(w+b\right) using the obtained values.

$$\left(w-7\right)\left(w+2\right)$$

To find equation solutions, solve w-7=0 and w+2=0.

$$w=7 w=-2$$

Variable w cannot be equal to -2.

$$w=7$$

Steps Using Factoring By Grouping

Variable w cannot be equal to any of the values -3,-2 since division by zero is not defined. Multiply both sides of the equation by \left(w+2\right)\left(w+3\right), the least common multiple of w^{2}+5w+6,w+2,w+3.

$$3+\left(w+3\right)\left(w-1\right)=\left(w+2\right)\times 7$$

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

$$3+w^{2}+2w-3=\left(w+2\right)\times 7$$

Subtract 3 from 3 to get 0.

$$w^{2}+2w=\left(w+2\right)\times 7$$

Use the distributive property to multiply w+2 by 7.

$$w^{2}+2w=7w+14$$

Subtract 7w from both sides.

$$w^{2}+2w-7w=14$$

Combine 2w and -7w to get -5w.

$$w^{2}-5w=14$$

Subtract 14 from both sides.

$$w^{2}-5w-14=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as w^{2}+aw+bw-14. To find a and b, set up a system to be solved.

$$a+b=-5 ab=1\left(-14\right)=-14$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -14.

$$1,-14 2,-7$$

Calculate the sum for each pair.

$$1-14=-13 2-7=-5$$

The solution is the pair that gives sum -5.

$$a=-7 b=2$$

Rewrite w^{2}-5w-14 as \left(w^{2}-7w\right)+\left(2w-14\right).

$$\left(w^{2}-7w\right)+\left(2w-14\right)$$

Factor out w in the first and 2 in the second group.

$$w\left(w-7\right)+2\left(w-7\right)$$

Factor out common term w-7 by using distributive property.

$$\left(w-7\right)\left(w+2\right)$$

To find equation solutions, solve w-7=0 and w+2=0.

$$w=7 w=-2$$

Variable w cannot be equal to -2.

$$w=7$$

Steps Using the Quadratic Formula

Variable w cannot be equal to any of the values -3,-2 since division by zero is not defined. Multiply both sides of the equation by \left(w+2\right)\left(w+3\right), the least common multiple of w^{2}+5w+6,w+2,w+3.

$$3+\left(w+3\right)\left(w-1\right)=\left(w+2\right)\times 7$$

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

$$3+w^{2}+2w-3=\left(w+2\right)\times 7$$

Subtract 3 from 3 to get 0.

$$w^{2}+2w=\left(w+2\right)\times 7$$

Use the distributive property to multiply w+2 by 7.

$$w^{2}+2w=7w+14$$

Subtract 7w from both sides.

$$w^{2}+2w-7w=14$$

Combine 2w and -7w to get -5w.

$$w^{2}-5w=14$$

Subtract 14 from both sides.

$$w^{2}-5w-14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -5 for b, and -14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$w=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\left(-14\right)}}{2}$$

Square -5.

$$w=\frac{-\left(-5\right)±\sqrt{25-4\left(-14\right)}}{2}$$

Multiply -4 times -14.

$$w=\frac{-\left(-5\right)±\sqrt{25+56}}{2}$$

Add 25 to 56.

$$w=\frac{-\left(-5\right)±\sqrt{81}}{2}$$

Take the square root of 81.

$$w=\frac{-\left(-5\right)±9}{2}$$

The opposite of -5 is 5.

$$w=\frac{5±9}{2}$$

Now solve the equation w=\frac{5±9}{2} when ± is plus. Add 5 to 9.

$$w=\frac{14}{2}$$

Divide 14 by 2.

$$w=7$$

Now solve the equation w=\frac{5±9}{2} when ± is minus. Subtract 9 from 5.

$$w=-\frac{4}{2}$$

Divide -4 by 2.

$$w=-2$$

The equation is now solved.

$$w=7 w=-2$$

Variable w cannot be equal to -2.

$$w=7$$

Steps for Completing the Square

Variable w cannot be equal to any of the values -3,-2 since division by zero is not defined. Multiply both sides of the equation by \left(w+2\right)\left(w+3\right), the least common multiple of w^{2}+5w+6,w+2,w+3.

$$3+\left(w+3\right)\left(w-1\right)=\left(w+2\right)\times 7$$

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

$$3+w^{2}+2w-3=\left(w+2\right)\times 7$$

Subtract 3 from 3 to get 0.

$$w^{2}+2w=\left(w+2\right)\times 7$$

Use the distributive property to multiply w+2 by 7.

$$w^{2}+2w=7w+14$$

Subtract 7w from both sides.

$$w^{2}+2w-7w=14$$

Combine 2w and -7w to get -5w.

$$w^{2}-5w=14$$

Divide -5, the coefficient of the x term, by 2 to get -\frac{5}{2}. Then add the square of -\frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$w^{2}-5w+\left(-\frac{5}{2}\right)^{2}=14+\left(-\frac{5}{2}\right)^{2}$$

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$w^{2}-5w+\frac{25}{4}=14+\frac{25}{4}$$

Add 14 to \frac{25}{4}.

$$w^{2}-5w+\frac{25}{4}=\frac{81}{4}$$

Factor w^{2}-5w+\frac{25}{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}.

$$\left(w-\frac{5}{2}\right)^{2}=\frac{81}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(w-\frac{5}{2}\right)^{2}}=\sqrt{\frac{81}{4}}$$

Simplify.

$$w-\frac{5}{2}=\frac{9}{2} w-\frac{5}{2}=-\frac{9}{2}$$

Add \frac{5}{2} to both sides of the equation.

$$w=7 w=-2$$

Variable w cannot be equal to -2.

$$w=7$$