Calculus Examples

$12 x^{2} - 56 x + 49 = 0$

Step 1

Factor by grouping.

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Step 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 $a \cdot c = 12 \cdot 49 = 588$ and whose sum is $b = - 56$ .

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Step 1.1.1

Factor $- 56$ out of $- 56 x$ .

$12 x^{2} - 56 x + 49 = 0$

Step 1.1.2

Rewrite $- 56$ as $- 14$ plus $- 42$

$12 x^{2} + (- 14 - 42) x + 49 = 0$

Step 1.1.3

Apply the distributive property.

$12 x^{2} - 14 x - 42 x + 49 = 0$

Step 1.2

Factor out the greatest common factor from each group.

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Step 1.2.1

Group the first two terms and the last two terms.

$(12 x^{2} - 14 x) - 42 x + 49 = 0$

Step 1.2.2

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

$2 x (6 x - 7) - 7 (6 x - 7) = 0$

Step 1.3

Factor the polynomial by factoring out the greatest common factor, $6 x - 7$ .

$(6 x - 7) (2 x - 7) = 0$

Step 2

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

$6 x - 7 = 0$

$2 x - 7 = 0$

Step 3

Set $6 x - 7$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $6 x - 7$ equal to $0$ .

$6 x - 7 = 0$

Step 3.2

Solve $6 x - 7 = 0$ for $x$ .

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Step 3.2.1

Add $7$ to both sides of the equation.

$6 x = 7$

Step 3.2.2

Divide each term in $6 x = 7$ by $6$ and simplify.

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Step 3.2.2.1

Divide each term in $6 x = 7$ by $6$ .

$\frac{6 x}{6} = \frac{7}{6}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $6$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{7}{6}$

Step 3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{7}{6}$

Step 4

Set $2 x - 7$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $2 x - 7$ equal to $0$ .

$2 x - 7 = 0$

Step 4.2

Solve $2 x - 7 = 0$ for $x$ .

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Step 4.2.1

Add $7$ to both sides of the equation.

$2 x = 7$

Step 4.2.2

Divide each term in $2 x = 7$ by $2$ and simplify.

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Step 4.2.2.1

Divide each term in $2 x = 7$ by $2$ .

$\frac{2 x}{2} = \frac{7}{2}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{7}{2}$

Step 4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{7}{2}$

Step 5

The final solution is all the values that make $(6 x - 7) (2 x - 7) = 0$ true.

$x = \frac{7}{6}, \frac{7}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{7}{6}, \frac{7}{2}$

Decimal Form:

$x = 1.1\overline{6}, 3.5$

Mixed Number Form:

$x = 1 \frac{1}{6}, 3 \frac{1}{2}$