Pre-Algebra Examples

$s^{2} = 9 + s (1 - 2 s)$

Step 1

Since $s$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$9 + s (1 - 2 s) = s^{2}$

Step 2

Simplify each term.

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Step 2.1

Apply the distributive property.

$9 + s \cdot 1 + s (- 2 s) = s^{2}$

Step 2.2

Multiply $s$ by $1$ .

$9 + s + s (- 2 s) = s^{2}$

Step 2.3

Rewrite using the commutative property of multiplication.

$9 + s - 2 s \cdot s = s^{2}$

Step 2.4

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

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Step 2.4.1

Move $s$ .

$9 + s - 2 (s \cdot s) = s^{2}$

Step 2.4.2

Multiply $s$ by $s$ .

$9 + s - 2 s^{2} = s^{2}$

Step 3

Move all terms containing $s$ to the left side of the equation.

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Step 3.1

Subtract $s^{2}$ from both sides of the equation.

$9 + s - 2 s^{2} - s^{2} = 0$

Step 3.2

Subtract $s^{2}$ from $- 2 s^{2}$ .

$9 + s - 3 s^{2} = 0$

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 $a = - 3$ , $b = 1$ , and $c = 9$ into the quadratic formula and solve for $s$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (- 3 \cdot 9)}}{2 \cdot - 3}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

One to any power is one.

$s = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 3 \cdot 9}}{2 \cdot - 3}$

Step 6.1.2

Multiply $- 4 \cdot - 3 \cdot 9$ .

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Step 6.1.2.1

Multiply $- 4$ by $- 3$ .

$s = \frac{- 1 \pm \sqrt{1 + 12 \cdot 9}}{2 \cdot - 3}$

Step 6.1.2.2

Multiply $12$ by $9$ .

$s = \frac{- 1 \pm \sqrt{1 + 108}}{2 \cdot - 3}$

Step 6.1.3

Add $1$ and $108$ .

$s = \frac{- 1 \pm \sqrt{109}}{2 \cdot - 3}$

Step 6.2

Multiply $2$ by $- 3$ .

$s = \frac{- 1 \pm \sqrt{109}}{- 6}$

Step 6.3

Simplify $\frac{- 1 \pm \sqrt{109}}{- 6}$ .

$s = \frac{1 \pm \sqrt{109}}{6}$

Step 7

The final answer is the combination of both solutions.

$s = \frac{1 + \sqrt{109}}{6}, \frac{1 - \sqrt{109}}{6}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$s = \frac{1 + \sqrt{109}}{6}, \frac{1 - \sqrt{109}}{6}$

Decimal Form:

$s = 1.90671775 \dots, - 1.57338441 \dots$