Trigonometry Examples

$r^{2} + 3 r = 7$

Step 1

Subtract $7$ from both sides of the equation.

$r^{2} + 3 r - 7 = 0$

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 $a = 1$ , $b = 3$ , and $c = - 7$ into the quadratic formula and solve for $r$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (1 \cdot - 7)}}{2 \cdot 1}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

$r = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 7}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4 \cdot 1 \cdot - 7$ .

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Step 4.1.2.1

Multiply $- 4$ by $1$ .

$r = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 7}}{2 \cdot 1}$

Step 4.1.2.2

Multiply $- 4$ by $- 7$ .

$r = \frac{- 3 \pm \sqrt{9 + 28}}{2 \cdot 1}$

Step 4.1.3

Add $9$ and $28$ .

$r = \frac{- 3 \pm \sqrt{37}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$r = \frac{- 3 \pm \sqrt{37}}{2}$

Step 5

The final answer is the combination of both solutions.

$r = - \frac{3 - \sqrt{37}}{2}, - \frac{3 + \sqrt{37}}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$r = - \frac{3 - \sqrt{37}}{2}, - \frac{3 + \sqrt{37}}{2}$

Decimal Form:

$r = 1.54138126 \dots, - 4.54138126 \dots$