Solve Using the Quadratic Formula (3w+4)(2w-7)=0

$(3 w + 4) (2 w - 7) = 0$

Step 1

Simplify the left side.

Simplify $(3 w + 4) (2 w - 7)$ .

Expand $(3 w + 4) (2 w - 7)$ using the FOIL Method.

Apply the distributive property.

$3 w (2 w - 7) + 4 (2 w - 7) = 0$

Apply the distributive property.

$3 w (2 w) + 3 w \cdot - 7 + 4 (2 w - 7) = 0$

Apply the distributive property.

$3 w (2 w) + 3 w \cdot - 7 + 4 (2 w) + 4 \cdot - 7 = 0$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$3 \cdot 2 w \cdot w + 3 w \cdot - 7 + 4 (2 w) + 4 \cdot - 7 = 0$

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

Move $w$ .

$3 \cdot 2 (w \cdot w) + 3 w \cdot - 7 + 4 (2 w) + 4 \cdot - 7 = 0$

Multiply $w$ by $w$ .

$3 \cdot 2 w^{2} + 3 w \cdot - 7 + 4 (2 w) + 4 \cdot - 7 = 0$

Multiply $3$ by $2$ .

$6 w^{2} + 3 w \cdot - 7 + 4 (2 w) + 4 \cdot - 7 = 0$

Multiply $- 7$ by $3$ .

$6 w^{2} - 21 w + 4 (2 w) + 4 \cdot - 7 = 0$

Multiply $2$ by $4$ .

$6 w^{2} - 21 w + 8 w + 4 \cdot - 7 = 0$

Multiply $4$ by $- 7$ .

$6 w^{2} - 21 w + 8 w - 28 = 0$

Add $- 21 w$ and $8 w$ .

$6 w^{2} - 13 w - 28 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 6$ , $b = - 13$ , and $c = - 28$ into the quadratic formula and solve for $w$ .

$\frac{13 \pm \sqrt{{(- 13)}^{2} - 4 \cdot (6 \cdot - 28)}}{2 \cdot 6}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 13$ to the power of $2$ .

$w = \frac{13 \pm \sqrt{169 - 4 \cdot 6 \cdot - 28}}{2 \cdot 6}$

Multiply $- 4 \cdot 6 \cdot - 28$ .

Multiply $- 4$ by $6$ .

$w = \frac{13 \pm \sqrt{169 - 24 \cdot - 28}}{2 \cdot 6}$

Multiply $- 24$ by $- 28$ .

$w = \frac{13 \pm \sqrt{169 + 672}}{2 \cdot 6}$

Add $169$ and $672$ .

$w = \frac{13 \pm \sqrt{841}}{2 \cdot 6}$

Rewrite $841$ as $29^{2}$ .

$w = \frac{13 \pm \sqrt{29^{2}}}{2 \cdot 6}$

Pull terms out from under the radical, assuming positive real numbers.

$w = \frac{13 \pm 29}{2 \cdot 6}$

Multiply $2$ by $6$ .

$w = \frac{13 \pm 29}{12}$

Step 5

The final answer is the combination of both solutions.

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

Solve Using the Quadratic Formula (3w+4)(2w-7)=0

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