Trigonometry Examples

$x^{2} + 4 x = \frac{9}{4}$

Step 1

Subtract $\frac{9}{4}$ from both sides of the equation.

$x^{2} + 4 x - \frac{9}{4} = 0$

Step 2

Multiply through by the least common denominator $4$ , then simplify.

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Step 2.1

Apply the distributive property.

$4 x^{2} + 4 (4 x) + 4 (- \frac{9}{4}) = 0$

Step 2.2

Simplify.

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Step 2.2.1

Multiply $4$ by $4$ .

$4 x^{2} + 16 x + 4 (- \frac{9}{4}) = 0$

Step 2.2.2

Cancel the common factor of $4$ .

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Step 2.2.2.1

Move the leading negative in $- \frac{9}{4}$ into the numerator.

$4 x^{2} + 16 x + 4 (\frac{- 9}{4}) = 0$

Step 2.2.2.2

Cancel the common factor.

$4 x^{2} + 16 x + 4 (\frac{- 9}{4}) = 0$

Step 2.2.2.3

Rewrite the expression.

$4 x^{2} + 16 x - 9 = 0$

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 $a = 4$ , $b = 16$ , and $c = - 9$ into the quadratic formula and solve for $x$ .

$\frac{- 16 \pm \sqrt{16^{2} - 4 \cdot (4 \cdot - 9)}}{2 \cdot 4}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $16$ to the power of $2$ .

$x = \frac{- 16 \pm \sqrt{256 - 4 \cdot 4 \cdot - 9}}{2 \cdot 4}$

Step 5.1.2

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

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Step 5.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{- 16 \pm \sqrt{256 - 16 \cdot - 9}}{2 \cdot 4}$

Step 5.1.2.2

Multiply $- 16$ by $- 9$ .

$x = \frac{- 16 \pm \sqrt{256 + 144}}{2 \cdot 4}$

Step 5.1.3

Add $256$ and $144$ .

$x = \frac{- 16 \pm \sqrt{400}}{2 \cdot 4}$

Step 5.1.4

Rewrite $400$ as $20^{2}$ .

$x = \frac{- 16 \pm \sqrt{20^{2}}}{2 \cdot 4}$

Step 5.1.5

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 16 \pm 20}{2 \cdot 4}$

Step 5.2

Multiply $2$ by $4$ .

$x = \frac{- 16 \pm 20}{8}$

Step 5.3

Simplify $\frac{- 16 \pm 20}{8}$ .

$x = \frac{- 4 \pm 5}{2}$

Step 6

The final answer is the combination of both solutions.

$x = \frac{1}{2}, - \frac{9}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{2}, - \frac{9}{2}$

Decimal Form:

$x = 0.5, - 4.5$

Mixed Number Form:

$x = \frac{1}{2}, - 4 \frac{1}{2}$