Finite Math Examples

$((w) (2 w - 10)) + (3 + w) = 256$

Step 1

Simplify $(w) (2 w - 10) + 3 + w$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$w (2 w) + w \cdot - 10 + 3 + w = 256$

Step 1.1.2

Rewrite using the commutative property of multiplication.

$2 w \cdot w + w \cdot - 10 + 3 + w = 256$

Step 1.1.3

Move $- 10$ to the left of $w$ .

$2 w \cdot w - 10 \cdot w + 3 + w = 256$

Step 1.1.4

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

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Step 1.1.4.1

Move $w$ .

$2 (w \cdot w) - 10 \cdot w + 3 + w = 256$

Step 1.1.4.2

Multiply $w$ by $w$ .

$2 w^{2} - 10 \cdot w + 3 + w = 256$

$2 w^{2} - 10 w + 3 + w = 256$

Step 1.2

Add $- 10 w$ and $w$ .

$2 w^{2} - 9 w + 3 = 256$

Step 2

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

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Step 2.1

Subtract $256$ from both sides of the equation.

$2 w^{2} - 9 w + 3 - 256 = 0$

Step 2.2

Subtract $256$ from $3$ .

$2 w^{2} - 9 w - 253 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 2$ , $b = - 9$ , and $c = - 253$ into the quadratic formula and solve for $w$ .

$\frac{9 \pm \sqrt{{(- 9)}^{2} - 4 \cdot (2 \cdot - 253)}}{2 \cdot 2}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 9$ to the power of $2$ .

$w = \frac{9 \pm \sqrt{81 - 4 \cdot 2 \cdot - 253}}{2 \cdot 2}$

Step 5.1.2

Multiply $- 4 \cdot 2 \cdot - 253$ .

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Step 5.1.2.1

Multiply $- 4$ by $2$ .

$w = \frac{9 \pm \sqrt{81 - 8 \cdot - 253}}{2 \cdot 2}$

Step 5.1.2.2

Multiply $- 8$ by $- 253$ .

$w = \frac{9 \pm \sqrt{81 + 2024}}{2 \cdot 2}$

Step 5.1.3

Add $81$ and $2024$ .

$w = \frac{9 \pm \sqrt{2105}}{2 \cdot 2}$

Step 5.2

Multiply $2$ by $2$ .

$w = \frac{9 \pm \sqrt{2105}}{4}$

Step 6

The final answer is the combination of both solutions.

$w = \frac{9 + \sqrt{2105}}{4}, \frac{9 - \sqrt{2105}}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$w = \frac{9 + \sqrt{2105}}{4}, \frac{9 - \sqrt{2105}}{4}$

Decimal Form:

$w = 13.72006974 \dots, - 9.22006974 \dots$