Calculus Examples

$3 x^{2} + 6 x - 24 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor $3$ out of $3 x^{2} + 6 x - 24$ .

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Step 1.1.1

Factor $3$ out of $3 x^{2}$ .

$3 (x^{2}) + 6 x - 24 = 0$

Step 1.1.2

Factor $3$ out of $6 x$ .

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

Step 1.1.3

Factor $3$ out of $- 24$ .

$3 x^{2} + 3 (2 x) + 3 \cdot - 8 = 0$

Step 1.1.4

Factor $3$ out of $3 x^{2} + 3 (2 x)$ .

$3 (x^{2} + 2 x) + 3 \cdot - 8 = 0$

Step 1.1.5

Factor $3$ out of $3 (x^{2} + 2 x) + 3 \cdot - 8$ .

$3 (x^{2} + 2 x - 8) = 0$

Step 1.2

Factor.

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Step 1.2.1

Factor $x^{2} + 2 x - 8$ using the AC method.

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Step 1.2.1.1

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 $- 8$ and whose sum is $2$ .

$- 2, 4$

Step 1.2.1.2

Write the factored form using these integers.

$3 ((x - 2) (x + 4)) = 0$

Step 1.2.2

Remove unnecessary parentheses.

$3 (x - 2) (x + 4) = 0$

Step 2

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

$x - 2 = 0$

$x + 4 = 0$

Step 3

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 3.2

Add $2$ to both sides of the equation.

$x = 2$

Step 4

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 4.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 5

The final solution is all the values that make $3 (x - 2) (x + 4) = 0$ true.

$x = 2, - 4$