Calculus Examples

$3 x^{2} - 12 x + 9 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor $3$ out of $3 x^{2} - 12 x + 9$ .

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Step 1.1.1

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

$3 (x^{2}) - 12 x + 9 = 0$

Step 1.1.2

Factor $3$ out of $- 12 x$ .

$3 (x^{2}) + 3 (- 4 x) + 9 = 0$

Step 1.1.3

Factor $3$ out of $9$ .

$3 x^{2} + 3 (- 4 x) + 3 \cdot 3 = 0$

Step 1.1.4

Factor $3$ out of $3 x^{2} + 3 (- 4 x)$ .

$3 (x^{2} - 4 x) + 3 \cdot 3 = 0$

Step 1.1.5

Factor $3$ out of $3 (x^{2} - 4 x) + 3 \cdot 3$ .

$3 (x^{2} - 4 x + 3) = 0$

Step 1.2

Factor.

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Step 1.2.1

Factor $x^{2} - 4 x + 3$ 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 $3$ and whose sum is $- 4$ .

$- 3, - 1$

Step 1.2.1.2

Write the factored form using these integers.

$3 ((x - 3) (x - 1)) = 0$

Step 1.2.2

Remove unnecessary parentheses.

$3 (x - 3) (x - 1) = 0$

Step 2

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

$x - 3 = 0$

$x - 1 = 0$

Step 3

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

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Step 3.1

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

$x - 3 = 0$

Step 3.2

Add $3$ to both sides of the equation.

$x = 3$

Step 4

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 4.2

Add $1$ to both sides of the equation.

$x = 1$

Step 5

The final solution is all the values that make $3 (x - 3) (x - 1) = 0$ true.

$x = 3, 1$