Trigonometry Examples

$- 1 x^{2} + 11 x - 24 = 0$

Step 1

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$- x^{2} + 11 x - 24 = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Factor $- 1$ out of $- x^{2} + 11 x - 24$ .

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Step 2.1.1

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

$- (x^{2}) + 11 x - 24 = 0$

Step 2.1.2

Factor $- 1$ out of $11 x$ .

$- (x^{2}) - (- 11 x) - 24 = 0$

Step 2.1.3

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

$- (x^{2}) - (- 11 x) - 1 \cdot 24 = 0$

Step 2.1.4

Factor $- 1$ out of $- (x^{2}) - (- 11 x)$ .

$- (x^{2} - 11 x) - 1 \cdot 24 = 0$

Step 2.1.5

Factor $- 1$ out of $- (x^{2} - 11 x) - 1 (24)$ .

$- (x^{2} - 11 x + 24) = 0$

Step 2.2

Factor.

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Step 2.2.1

Factor $x^{2} - 11 x + 24$ using the AC method.

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Step 2.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $24$ and whose sum is $- 11$ .

$- 8, - 3$

Step 2.2.1.2

Write the factored form using these integers.

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

Step 2.2.2

Remove unnecessary parentheses.

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

Step 3

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

$x - 8 = 0$

$x - 3 = 0$

Step 4

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

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Step 4.1

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

$x - 8 = 0$

Step 4.2

Add $8$ to both sides of the equation.

$x = 8$

Step 5

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 5.2

Add $3$ to both sides of the equation.

$x = 3$

Step 6

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

$x = 8, 3$