Trigonometry Examples

$13 x = 2 x^{2} - 24$

Step 1

Subtract $2 x^{2}$ from both sides of the equation.

$13 x - 2 x^{2} = - 24$

Step 2

Add $24$ to both sides of the equation.

$13 x - 2 x^{2} + 24 = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Factor $- 1$ out of $13 x - 2 x^{2} + 24$ .

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Step 3.1.1

Reorder $13 x$ and $- 2 x^{2}$ .

$- 2 x^{2} + 13 x + 24 = 0$

Step 3.1.2

Factor $- 1$ out of $- 2 x^{2}$ .

$- (2 x^{2}) + 13 x + 24 = 0$

Step 3.1.3

Factor $- 1$ out of $13 x$ .

$- (2 x^{2}) - (- 13 x) + 24 = 0$

Step 3.1.4

Rewrite $24$ as $- 1 (- 24)$ .

$- (2 x^{2}) - (- 13 x) - 1 \cdot - 24 = 0$

Step 3.1.5

Factor $- 1$ out of $- (2 x^{2}) - (- 13 x)$ .

$- (2 x^{2} - 13 x) - 1 \cdot - 24 = 0$

Step 3.1.6

Factor $- 1$ out of $- (2 x^{2} - 13 x) - 1 (- 24)$ .

$- (2 x^{2} - 13 x - 24) = 0$

Step 3.2

Factor.

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Step 3.2.1

Factor by grouping.

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Step 3.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 24 = - 48$ and whose sum is $b = - 13$ .

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Step 3.2.1.1.1

Factor $- 13$ out of $- 13 x$ .

$- (2 x^{2} - 13 x - 24) = 0$

Step 3.2.1.1.2

Rewrite $- 13$ as $3$ plus $- 16$

$- (2 x^{2} + (3 - 16) x - 24) = 0$

Step 3.2.1.1.3

Apply the distributive property.

$- (2 x^{2} + 3 x - 16 x - 24) = 0$

Step 3.2.1.2

Factor out the greatest common factor from each group.

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Step 3.2.1.2.1

Group the first two terms and the last two terms.

$- ((2 x^{2} + 3 x) - 16 x - 24) = 0$

Step 3.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$- (x (2 x + 3) - 8 (2 x + 3)) = 0$

Step 3.2.1.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 3$ .

$- ((2 x + 3) (x - 8)) = 0$

Step 3.2.2

Remove unnecessary parentheses.

$- (2 x + 3) (x - 8) = 0$

Step 4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$2 x + 3 = 0$

$x - 8 = 0$

Step 5

Set $2 x + 3$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $2 x + 3$ equal to $0$ .

$2 x + 3 = 0$

Step 5.2

Solve $2 x + 3 = 0$ for $x$ .

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Step 5.2.1

Subtract $3$ from both sides of the equation.

$2 x = - 3$

Step 5.2.2

Divide each term in $2 x = - 3$ by $2$ and simplify.

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Step 5.2.2.1

Divide each term in $2 x = - 3$ by $2$ .

$\frac{2 x}{2} = \frac{- 3}{2}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 3}{2}$

Step 5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{2}$

Step 5.2.2.3

Simplify the right side.

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Step 5.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{3}{2}$

Step 6

Set $x - 8$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Step 6.2

Add $8$ to both sides of the equation.

$x = 8$

Step 7

The final solution is all the values that make $- (2 x + 3) (x - 8) = 0$ true.

$x = - \frac{3}{2}, 8$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{3}{2}, 8$

Decimal Form:

$x = - 1.5, 8$

Mixed Number Form:

$x = - 1 \frac{1}{2}, 8$