Solve 3w(w+8)+w(w-4)/2-3=5-w^2

$$\frac { 3 w ( w + 8 ) + w ( w - 4 ) } { 2 } - 3 = 5 - w ^ { 2 }$$

Solve for w

$$w=-4$$

$$w=\frac{2}{3}\approx 0.666666667$$

Steps Using Factoring By Grouping

Multiply both sides of the equation by 2.

$$3w\left(w+8\right)+w\left(w-4\right)-6=10-2w^{2}$$

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

$$3w^{2}+24w+w\left(w-4\right)-6=10-2w^{2}$$

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

$$3w^{2}+24w+w^{2}-4w-6=10-2w^{2}$$

Combine 3w^{2} and w^{2} to get 4w^{2}.

$$4w^{2}+24w-4w-6=10-2w^{2}$$

Combine 24w and -4w to get 20w.

$$4w^{2}+20w-6=10-2w^{2}$$

Subtract 10 from both sides.

$$4w^{2}+20w-6-10=-2w^{2}$$

Subtract 10 from -6 to get -16.

$$4w^{2}+20w-16=-2w^{2}$$

Add 2w^{2} to both sides.

$$4w^{2}+20w-16+2w^{2}=0$$

Combine 4w^{2} and 2w^{2} to get 6w^{2}.

$$6w^{2}+20w-16=0$$

Divide both sides by 2.

$$3w^{2}+10w-8=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 3w^{2}+aw+bw-8. To find a and b, set up a system to be solved.

$$a+b=10 ab=3\left(-8\right)=-24$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -24.

$$-1,24 -2,12 -3,8 -4,6$$

Calculate the sum for each pair.

$$-1+24=23 -2+12=10 -3+8=5 -4+6=2$$

The solution is the pair that gives sum 10.

$$a=-2 b=12$$

Rewrite 3w^{2}+10w-8 as \left(3w^{2}-2w\right)+\left(12w-8\right).

$$\left(3w^{2}-2w\right)+\left(12w-8\right)$$

Factor out w in the first and 4 in the second group.

$$w\left(3w-2\right)+4\left(3w-2\right)$$

Factor out common term 3w-2 by using distributive property.

$$\left(3w-2\right)\left(w+4\right)$$

To find equation solutions, solve 3w-2=0 and w+4=0.

$$w=\frac{2}{3} w=-4$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$3w\left(w+8\right)+w\left(w-4\right)-6=10-2w^{2}$$

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

$$3w^{2}+24w+w\left(w-4\right)-6=10-2w^{2}$$

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

$$3w^{2}+24w+w^{2}-4w-6=10-2w^{2}$$

Combine 3w^{2} and w^{2} to get 4w^{2}.

$$4w^{2}+24w-4w-6=10-2w^{2}$$

Combine 24w and -4w to get 20w.

$$4w^{2}+20w-6=10-2w^{2}$$

Subtract 10 from both sides.

$$4w^{2}+20w-6-10=-2w^{2}$$

Subtract 10 from -6 to get -16.

$$4w^{2}+20w-16=-2w^{2}$$

Add 2w^{2} to both sides.

$$4w^{2}+20w-16+2w^{2}=0$$

Combine 4w^{2} and 2w^{2} to get 6w^{2}.

$$6w^{2}+20w-16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 20 for b, and -16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$w=\frac{-20±\sqrt{20^{2}-4\times 6\left(-16\right)}}{2\times 6}$$

Square 20.

$$w=\frac{-20±\sqrt{400-4\times 6\left(-16\right)}}{2\times 6}$$

Multiply -4 times 6.

$$w=\frac{-20±\sqrt{400-24\left(-16\right)}}{2\times 6}$$

Multiply -24 times -16.

$$w=\frac{-20±\sqrt{400+384}}{2\times 6}$$

Add 400 to 384.

$$w=\frac{-20±\sqrt{784}}{2\times 6}$$

Take the square root of 784.

$$w=\frac{-20±28}{2\times 6}$$

Multiply 2 times 6.

$$w=\frac{-20±28}{12}$$

Now solve the equation w=\frac{-20±28}{12} when ± is plus. Add -20 to 28.

$$w=\frac{8}{12}$$

Reduce the fraction \frac{8}{12} to lowest terms by extracting and canceling out 4.

$$w=\frac{2}{3}$$

Now solve the equation w=\frac{-20±28}{12} when ± is minus. Subtract 28 from -20.

$$w=-\frac{48}{12}$$

Divide -48 by 12.

$$w=-4$$

The equation is now solved.

$$w=\frac{2}{3} w=-4$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$3w\left(w+8\right)+w\left(w-4\right)-6=10-2w^{2}$$

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

$$3w^{2}+24w+w\left(w-4\right)-6=10-2w^{2}$$

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

$$3w^{2}+24w+w^{2}-4w-6=10-2w^{2}$$

Combine 3w^{2} and w^{2} to get 4w^{2}.

$$4w^{2}+24w-4w-6=10-2w^{2}$$

Combine 24w and -4w to get 20w.

$$4w^{2}+20w-6=10-2w^{2}$$

Add 2w^{2} to both sides.

$$4w^{2}+20w-6+2w^{2}=10$$

Combine 4w^{2} and 2w^{2} to get 6w^{2}.

$$6w^{2}+20w-6=10$$

Add 6 to both sides.

$$6w^{2}+20w=10+6$$

Add 10 and 6 to get 16.

$$6w^{2}+20w=16$$

Divide both sides by 6.

$$\frac{6w^{2}+20w}{6}=\frac{16}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$w^{2}+\frac{20}{6}w=\frac{16}{6}$$

Reduce the fraction \frac{20}{6} to lowest terms by extracting and canceling out 2.

$$w^{2}+\frac{10}{3}w=\frac{16}{6}$$

Reduce the fraction \frac{16}{6} to lowest terms by extracting and canceling out 2.

$$w^{2}+\frac{10}{3}w=\frac{8}{3}$$

Divide \frac{10}{3}, the coefficient of the x term, by 2 to get \frac{5}{3}. Then add the square of \frac{5}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$w^{2}+\frac{10}{3}w+\left(\frac{5}{3}\right)^{2}=\frac{8}{3}+\left(\frac{5}{3}\right)^{2}$$

Square \frac{5}{3} by squaring both the numerator and the denominator of the fraction.

$$w^{2}+\frac{10}{3}w+\frac{25}{9}=\frac{8}{3}+\frac{25}{9}$$

Add \frac{8}{3} to \frac{25}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$w^{2}+\frac{10}{3}w+\frac{25}{9}=\frac{49}{9}$$

Factor w^{2}+\frac{10}{3}w+\frac{25}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(w+\frac{5}{3}\right)^{2}=\frac{49}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(w+\frac{5}{3}\right)^{2}}=\sqrt{\frac{49}{9}}$$

Simplify.

$$w+\frac{5}{3}=\frac{7}{3} w+\frac{5}{3}=-\frac{7}{3}$$

Subtract \frac{5}{3} from both sides of the equation.

$$w=\frac{2}{3} w=-4$$