Solve for w w*0.5w+9z=2z+3w

$w \cdot 0.5 w + 9 z = 2 z + 3 w$

Step 1

Simplify each term.

Rewrite using the commutative property of multiplication.

$0.5 w \cdot w + 9 z = 2 z + 3 w$

Multiply $w$ by $w$ by adding the exponents.

Move $w$ .

$0.5 (w \cdot w) + 9 z = 2 z + 3 w$

Multiply $w$ by $w$ .

$0.5 w^{2} + 9 z = 2 z + 3 w$

Step 2

Subtract $3 w$ from both sides of the equation.

$0.5 w^{2} + 9 z - 3 w = 2 z$

Step 3

Move all terms to the left side of the equation and simplify.

Subtract $2 z$ from both sides of the equation.

$0.5 w^{2} + 9 z - 3 w - 2 z = 0$

Subtract $2 z$ from $9 z$ .

$0.5 w^{2} - 3 w + 7 z = 0$

Step 4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 5

Substitute the values $a = 0.5$ , $b = - 3$ , and $c = 7 z$ into the quadratic formula and solve for $w$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (0.5 \cdot (7 z))}}{2 \cdot 0.5}$

Step 6

Simplify.

Simplify the numerator.

Raise $- 3$ to the power of $2$ .

$w = \frac{3 \pm \sqrt{9 - 4 \cdot 0.5 \cdot (7 z)}}{2 \cdot 0.5}$

Multiply $- 4 \cdot 0.5 \cdot 7$ .

Multiply $- 4$ by $0.5$ .

$w = \frac{3 \pm \sqrt{9 - 2 \cdot (7 z)}}{2 \cdot 0.5}$

Multiply $- 2$ by $7$ .

$w = \frac{3 \pm \sqrt{9 - 14 z}}{2 \cdot 0.5}$

Factor $1$ out of $9 - 14 z$ .

Rewrite $9$ as $1 (9)$ .

$w = \frac{3 \pm \sqrt{1 (9) - 14 z}}{2 \cdot 0.5}$

Factor $1$ out of $- 14 z$ .

$w = \frac{3 \pm \sqrt{1 (9) + 1 (- 14 z)}}{2 \cdot 0.5}$

Factor $1$ out of $1 (9) + 1 (- 14 z)$ .

$w = \frac{3 \pm \sqrt{1 (9 - 14 z)}}{2 \cdot 0.5}$

Multiply $9 - 14 z$ by $1$ .

$w = \frac{3 \pm \sqrt{9 - 14 z}}{2 \cdot 0.5}$

Multiply $2$ by $0.5$ .

$w = \frac{3 \pm \sqrt{9 - 14 z}}{1}$

Divide $3 \pm \sqrt{9 - 14 z}$ by $1$ .

$w = 3 \pm \sqrt{9 - 14 z}$

Step 7

The final answer is the combination of both solutions.

$w = 3 + \sqrt{9 - 14 z}$

$w = 3 - \sqrt{9 - 14 z}$

Solve for w w*0.5w+9z=2z+3w

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