Calculus Examples

$0 = 12 x^{2} - 176 x + 484$

Step 1

Rewrite the equation as $12 x^{2} - 176 x + 484 = 0$ .

$12 x^{2} - 176 x + 484 = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Factor $4$ out of $12 x^{2} - 176 x + 484$ .

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Step 2.1.1

Factor $4$ out of $12 x^{2}$ .

$4 (3 x^{2}) - 176 x + 484 = 0$

Step 2.1.2

Factor $4$ out of $- 176 x$ .

$4 (3 x^{2}) + 4 (- 44 x) + 484 = 0$

Step 2.1.3

Factor $4$ out of $484$ .

$4 (3 x^{2}) + 4 (- 44 x) + 4 (121) = 0$

Step 2.1.4

Factor $4$ out of $4 (3 x^{2}) + 4 (- 44 x)$ .

$4 (3 x^{2} - 44 x) + 4 (121) = 0$

Step 2.1.5

Factor $4$ out of $4 (3 x^{2} - 44 x) + 4 (121)$ .

$4 (3 x^{2} - 44 x + 121) = 0$

Step 2.2

Factor.

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Step 2.2.1

Factor by grouping.

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Step 2.2.1.1

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 $a \cdot c = 3 \cdot 121 = 363$ and whose sum is $b = - 44$ .

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Step 2.2.1.1.1

Factor $- 44$ out of $- 44 x$ .

$4 (3 x^{2} - 44 x + 121) = 0$

Step 2.2.1.1.2

Rewrite $- 44$ as $- 11$ plus $- 33$

$4 (3 x^{2} + (- 11 - 33) x + 121) = 0$

Step 2.2.1.1.3

Apply the distributive property.

$4 (3 x^{2} - 11 x - 33 x + 121) = 0$

Step 2.2.1.2

Factor out the greatest common factor from each group.

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Step 2.2.1.2.1

Group the first two terms and the last two terms.

$4 ((3 x^{2} - 11 x) - 33 x + 121) = 0$

Step 2.2.1.2.2

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

$4 (x (3 x - 11) - 11 (3 x - 11)) = 0$

Step 2.2.1.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 11$ .

$4 ((3 x - 11) (x - 11)) = 0$

Step 2.2.2

Remove unnecessary parentheses.

$4 (3 x - 11) (x - 11) = 0$

Step 3

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

$3 x - 11 = 0$

$x - 11 = 0$

Step 4

Set $3 x - 11$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $3 x - 11$ equal to $0$ .

$3 x - 11 = 0$

Step 4.2

Solve $3 x - 11 = 0$ for $x$ .

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Step 4.2.1

Add $11$ to both sides of the equation.

$3 x = 11$

Step 4.2.2

Divide each term in $3 x = 11$ by $3$ and simplify.

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Step 4.2.2.1

Divide each term in $3 x = 11$ by $3$ .

$\frac{3 x}{3} = \frac{11}{3}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $3$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{11}{3}$

Step 4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{11}{3}$

Step 5

Set $x - 11$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x - 11$ equal to $0$ .

$x - 11 = 0$

Step 5.2

Add $11$ to both sides of the equation.

$x = 11$

Step 6

The final solution is all the values that make $4 (3 x - 11) (x - 11) = 0$ true.

$x = \frac{11}{3}, 11$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{11}{3}, 11$

Decimal Form:

$x = 3.\overline{6}, 11$

Mixed Number Form:

$x = 3 \frac{2}{3}, 11$