Calculus Examples

$12 x^{2} - 136 x + 289 = 0$

Step 1

Factor by grouping.

Tap for more steps...

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 $a \cdot c = 12 \cdot 289 = 3468$ and whose sum is $b = - 136$ .

Tap for more steps...

Step 1.1.1

Factor $- 136$ out of $- 136 x$ .

$12 x^{2} - 136 x + 289 = 0$

Step 1.1.2

Rewrite $- 136$ as $- 34$ plus $- 102$

$12 x^{2} + (- 34 - 102) x + 289 = 0$

Step 1.1.3

Apply the distributive property.

$12 x^{2} - 34 x - 102 x + 289 = 0$

Step 1.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 1.2.1

Group the first two terms and the last two terms.

$(12 x^{2} - 34 x) - 102 x + 289 = 0$

Step 1.2.2

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

$2 x (6 x - 17) - 17 (6 x - 17) = 0$

Step 1.3

Factor the polynomial by factoring out the greatest common factor, $6 x - 17$ .

$(6 x - 17) (2 x - 17) = 0$

Step 2

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

$6 x - 17 = 0$

$2 x - 17 = 0$

Step 3

Set $6 x - 17$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.1

Set $6 x - 17$ equal to $0$ .

$6 x - 17 = 0$

Step 3.2

Solve $6 x - 17 = 0$ for $x$ .

Tap for more steps...

Step 3.2.1

Add $17$ to both sides of the equation.

$6 x = 17$

Step 3.2.2

Divide each term in $6 x = 17$ by $6$ and simplify.

Tap for more steps...

Step 3.2.2.1

Divide each term in $6 x = 17$ by $6$ .

$\frac{6 x}{6} = \frac{17}{6}$

Step 3.2.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.2.1

Cancel the common factor of $6$ .

Tap for more steps...

Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{17}{6}$

Step 3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{17}{6}$

Step 4

Set $2 x - 17$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.1

Set $2 x - 17$ equal to $0$ .

$2 x - 17 = 0$

Step 4.2

Solve $2 x - 17 = 0$ for $x$ .

Tap for more steps...

Step 4.2.1

Add $17$ to both sides of the equation.

$2 x = 17$

Step 4.2.2

Divide each term in $2 x = 17$ by $2$ and simplify.

Tap for more steps...

Step 4.2.2.1

Divide each term in $2 x = 17$ by $2$ .

$\frac{2 x}{2} = \frac{17}{2}$

Step 4.2.2.2

Simplify the left side.

Tap for more steps...

Step 4.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{17}{2}$

Step 4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{17}{2}$

Step 5

The final solution is all the values that make $(6 x - 17) (2 x - 17) = 0$ true.

$x = \frac{17}{6}, \frac{17}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{17}{6}, \frac{17}{2}$

Decimal Form:

$x = 2.8\overline{3}, 8.5$

Mixed Number Form:

$x = 2 \frac{5}{6}, 8 \frac{1}{2}$