Calculus Examples

$0 = 3 q^{2} - 15 q + 18$

Step 1

Rewrite the equation as $3 q^{2} - 15 q + 18 = 0$ .

$3 q^{2} - 15 q + 18 = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Factor $3$ out of $3 q^{2} - 15 q + 18$ .

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Step 2.1.1

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

$3 (q^{2}) - 15 q + 18 = 0$

Step 2.1.2

Factor $3$ out of $- 15 q$ .

$3 (q^{2}) + 3 (- 5 q) + 18 = 0$

Step 2.1.3

Factor $3$ out of $18$ .

$3 q^{2} + 3 (- 5 q) + 3 \cdot 6 = 0$

Step 2.1.4

Factor $3$ out of $3 q^{2} + 3 (- 5 q)$ .

$3 (q^{2} - 5 q) + 3 \cdot 6 = 0$

Step 2.1.5

Factor $3$ out of $3 (q^{2} - 5 q) + 3 \cdot 6$ .

$3 (q^{2} - 5 q + 6) = 0$

Step 2.2

Factor.

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Step 2.2.1

Factor $q^{2} - 5 q + 6$ 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 $6$ and whose sum is $- 5$ .

$- 3, - 2$

Step 2.2.1.2

Write the factored form using these integers.

$3 ((q - 3) (q - 2)) = 0$

Step 2.2.2

Remove unnecessary parentheses.

$3 (q - 3) (q - 2) = 0$

Step 3

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

$q - 3 = 0$

$q - 2 = 0$

Step 4

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

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Step 4.1

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

$q - 3 = 0$

Step 4.2

Add $3$ to both sides of the equation.

$q = 3$

Step 5

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

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Step 5.1

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

$q - 2 = 0$

Step 5.2

Add $2$ to both sides of the equation.

$q = 2$

Step 6

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

$q = 3, 2$