Trigonometry Examples

$4 x^{2} - 20 x + 25 = 7$

Step 1

Subtract $7$ from both sides of the equation.

$4 x^{2} - 20 x + 25 - 7 = 0$

Step 2

Subtract $7$ from $25$ .

$4 x^{2} - 20 x + 18 = 0$

Step 3

Factor $2$ out of $4 x^{2} - 20 x + 18$ .

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Step 3.1

Factor $2$ out of $4 x^{2}$ .

$2 (2 x^{2}) - 20 x + 18 = 0$

Step 3.2

Factor $2$ out of $- 20 x$ .

$2 (2 x^{2}) + 2 (- 10 x) + 18 = 0$

Step 3.3

Factor $2$ out of $18$ .

$2 (2 x^{2}) + 2 (- 10 x) + 2 (9) = 0$

Step 3.4

Factor $2$ out of $2 (2 x^{2}) + 2 (- 10 x)$ .

$2 (2 x^{2} - 10 x) + 2 (9) = 0$

Step 3.5

Factor $2$ out of $2 (2 x^{2} - 10 x) + 2 (9)$ .

$2 (2 x^{2} - 10 x + 9) = 0$

Step 4

Divide each term in $2 (2 x^{2} - 10 x + 9) = 0$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 (2 x^{2} - 10 x + 9) = 0$ by $2$ .

$\frac{2 (2 x^{2} - 10 x + 9)}{2} = \frac{0}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 (2 x^{2} - 10 x + 9)}{2} = \frac{0}{2}$

Step 4.2.1.2

Divide $2 x^{2} - 10 x + 9$ by $1$ .

$2 x^{2} - 10 x + 9 = \frac{0}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $2$ .

$2 x^{2} - 10 x + 9 = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = 2$ , $b = - 10$ , and $c = 9$ into the quadratic formula and solve for $x$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (2 \cdot 9)}}{2 \cdot 2}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 2 \cdot 9}}{2 \cdot 2}$

Step 7.1.2

Multiply $- 4 \cdot 2 \cdot 9$ .

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Step 7.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{10 \pm \sqrt{100 - 8 \cdot 9}}{2 \cdot 2}$

Step 7.1.2.2

Multiply $- 8$ by $9$ .

$x = \frac{10 \pm \sqrt{100 - 72}}{2 \cdot 2}$

Step 7.1.3

Subtract $72$ from $100$ .

$x = \frac{10 \pm \sqrt{28}}{2 \cdot 2}$

Step 7.1.4

Rewrite $28$ as $2^{2} \cdot 7$ .

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Step 7.1.4.1

Factor $4$ out of $28$ .

$x = \frac{10 \pm \sqrt{4 (7)}}{2 \cdot 2}$

Step 7.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{10 \pm \sqrt{2^{2} \cdot 7}}{2 \cdot 2}$

Step 7.1.5

Pull terms out from under the radical.

$x = \frac{10 \pm 2 \sqrt{7}}{2 \cdot 2}$

Step 7.2

Multiply $2$ by $2$ .

$x = \frac{10 \pm 2 \sqrt{7}}{4}$

Step 7.3

Simplify $\frac{10 \pm 2 \sqrt{7}}{4}$ .

$x = \frac{5 \pm \sqrt{7}}{2}$

Step 8

The final answer is the combination of both solutions.

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

Step 9

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 3.82287565 \dots, 1.17712434 \dots$