Trigonometry Examples

$u^{2} - 7 u + 1 = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 1$ , $b = - 7$ , and $c = 1$ into the quadratic formula and solve for $u$ .

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

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $- 7$ to the power of $2$ .

$u = \frac{7 \pm \sqrt{49 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 3.1.2

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

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Step 3.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{7 \pm \sqrt{49 - 4 \cdot 1}}{2 \cdot 1}$

Step 3.1.2.2

Multiply $- 4$ by $1$ .

$u = \frac{7 \pm \sqrt{49 - 4}}{2 \cdot 1}$

Step 3.1.3

Subtract $4$ from $49$ .

$u = \frac{7 \pm \sqrt{45}}{2 \cdot 1}$

Step 3.1.4

Rewrite $45$ as $3^{2} \cdot 5$ .

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Step 3.1.4.1

Factor $9$ out of $45$ .

$u = \frac{7 \pm \sqrt{9 (5)}}{2 \cdot 1}$

Step 3.1.4.2

Rewrite $9$ as $3^{2}$ .

$u = \frac{7 \pm \sqrt{3^{2} \cdot 5}}{2 \cdot 1}$

Step 3.1.5

Pull terms out from under the radical.

$u = \frac{7 \pm 3 \sqrt{5}}{2 \cdot 1}$

Step 3.2

Multiply $2$ by $1$ .

$u = \frac{7 \pm 3 \sqrt{5}}{2}$

Step 4

The final answer is the combination of both solutions.

$u = \frac{7 + 3 \sqrt{5}}{2}, \frac{7 - 3 \sqrt{5}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$u = \frac{7 + 3 \sqrt{5}}{2}, \frac{7 - 3 \sqrt{5}}{2}$

Decimal Form:

$u = 6.85410196 \dots, 0.14589803 \dots$