Trigonometry Examples

$(x + 8) (x - 5) = 9$

Step 1

Simplify $(x + 8) (x - 5)$ .

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Step 1.1

Expand $(x + 8) (x - 5)$ using the FOIL Method.

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Step 1.1.1

Apply the distributive property.

$x (x - 5) + 8 (x - 5) = 9$

Step 1.1.2

Apply the distributive property.

$x \cdot x + x \cdot - 5 + 8 (x - 5) = 9$

Step 1.1.3

Apply the distributive property.

$x \cdot x + x \cdot - 5 + 8 x + 8 \cdot - 5 = 9$

Step 1.2

Simplify and combine like terms.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 5 + 8 x + 8 \cdot - 5 = 9$

Step 1.2.1.2

Move $- 5$ to the left of $x$ .

$x^{2} - 5 \cdot x + 8 x + 8 \cdot - 5 = 9$

Step 1.2.1.3

Multiply $8$ by $- 5$ .

$x^{2} - 5 x + 8 x - 40 = 9$

Step 1.2.2

Add $- 5 x$ and $8 x$ .

$x^{2} + 3 x - 40 = 9$

Step 2

Move all terms to the left side of the equation and simplify.

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Step 2.1

Subtract $9$ from both sides of the equation.

$x^{2} + 3 x - 40 - 9 = 0$

Step 2.2

Subtract $9$ from $- 40$ .

$x^{2} + 3 x - 49 = 0$

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 $a = 1$ , $b = 3$ , and $c = - 49$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (1 \cdot - 49)}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 49}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot - 49$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 49}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $- 49$ .

$x = \frac{- 3 \pm \sqrt{9 + 196}}{2 \cdot 1}$

Step 5.1.3

Add $9$ and $196$ .

$x = \frac{- 3 \pm \sqrt{205}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm \sqrt{205}}{2}$

Step 6

The final answer is the combination of both solutions.

$x = - \frac{3 - \sqrt{205}}{2}, - \frac{3 + \sqrt{205}}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{3 - \sqrt{205}}{2}, - \frac{3 + \sqrt{205}}{2}$

Decimal Form:

$x = 5.65891053 \dots, - 8.65891053 \dots$