Finite Math Examples

$s (10 s + 3) = 11$

Step 1

Simplify $s (10 s + 3)$ .

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Step 1.1

Simplify by multiplying through.

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Step 1.1.1

Apply the distributive property.

$s (10 s) + s \cdot 3 = 11$

Step 1.1.2

Reorder.

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Step 1.1.2.1

Rewrite using the commutative property of multiplication.

$10 s \cdot s + s \cdot 3 = 11$

Step 1.1.2.2

Move $3$ to the left of $s$ .

$10 s \cdot s + 3 \cdot s = 11$

Step 1.2

Multiply $s$ by $s$ by adding the exponents.

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Step 1.2.1

Move $s$ .

$10 (s \cdot s) + 3 \cdot s = 11$

Step 1.2.2

Multiply $s$ by $s$ .

$10 s^{2} + 3 \cdot s = 11$

$10 s^{2} + 3 s = 11$

Step 2

Subtract $11$ from both sides of the equation.

$10 s^{2} + 3 s - 11 = 0$

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 $a = 10$ , $b = 3$ , and $c = - 11$ into the quadratic formula and solve for $s$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (10 \cdot - 11)}}{2 \cdot 10}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$s = \frac{- 3 \pm \sqrt{9 - 4 \cdot 10 \cdot - 11}}{2 \cdot 10}$

Step 5.1.2

Multiply $- 4 \cdot 10 \cdot - 11$ .

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Step 5.1.2.1

Multiply $- 4$ by $10$ .

$s = \frac{- 3 \pm \sqrt{9 - 40 \cdot - 11}}{2 \cdot 10}$

Step 5.1.2.2

Multiply $- 40$ by $- 11$ .

$s = \frac{- 3 \pm \sqrt{9 + 440}}{2 \cdot 10}$

Step 5.1.3

Add $9$ and $440$ .

$s = \frac{- 3 \pm \sqrt{449}}{2 \cdot 10}$

Step 5.2

Multiply $2$ by $10$ .

$s = \frac{- 3 \pm \sqrt{449}}{20}$

Step 6

The final answer is the combination of both solutions.

$s = - \frac{3 - \sqrt{449}}{20}, - \frac{3 + \sqrt{449}}{20}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$s = - \frac{3 - \sqrt{449}}{20}, - \frac{3 + \sqrt{449}}{20}$

Decimal Form:

$s = 0.90948100 \dots, - 1.20948100 \dots$