Question

show all steps to complete 3x^2-13x-10

Answer 1

To complete the square for the quadratic expression \(3x^2 - 13x - 10\), we will follow these steps:

### Step 1: Factor out the leading coefficient from the first two terms.
The leading coefficient here is 3. We will factor it out from the first two terms:

\[
3x^2 - 13x - 10 = 3(x^2 - \frac{13}{3}x) - 10
\]

### Step 2: Complete the square.
Next, we need to complete the square inside the parentheses. To do this, we take the coefficient of \(x\) in the expression \(x^2 - \frac{13}{3}x\), which is \(-\frac{13}{3}\), divide it by 2, and then square it.

\[
\text{Coefficient of } x = -\frac{13}{3}
\]
\[
\text{Divide by 2: } -\frac{13}{3} \cdot \frac{1}{2} = -\frac{13}{6}
\]
\[
\text{Square it: } \left(-\frac{13}{6}\right)^2 = \frac{169}{36}
\]

### Step 3: Add and subtract the squared term inside the parentheses.
Now we will add and subtract \(\frac{169}{36}\) inside the parentheses:

\[
3\left(x^2 - \frac{13}{3}x + \frac{169}{36} - \frac{169}{36}\right) - 10
\]

Rearranging gives:

\[
3\left[\left(x - \frac{13}{6}\right)^2 - \frac{169}{36}\right] - 10
\]

### Step 4: Distribute the 3 and simplify.
Now we distribute the 3:

\[
3\left(x - \frac{13}{6}\right)^2 - 3 \cdot \frac{169}{36} - 10
\]

Calculating \(3 \cdot \frac{169}{36}\):

\[
3 \cdot \frac{169}{36} = \frac{507}{36}
\]

So the expression becomes:

\[
3\left(x - \frac{13}{6}\right)^2 - \frac{507}{36} - 10
\]

### Step 5: Convert -10 to have a denominator of 36.
We convert -10 to a fraction with a denominator of 36:

\[
-10 = -\frac{360}{36}
\]

### Step 6: Combine like terms.
Now, we combine the constant terms:

\[
3\left(x - \frac{13}{6}\right)^2 - \frac{507}{36} - \frac{360}{36}
\]

This simplifies to:

\[
3\left(x - \frac{13}{6}\right)^2 - \frac{867}{36}
\]

### Final Result
Thus, the expression \(3x^2 - 13x - 10\) in vertex form is:

\[
3\left(x - \frac{13}{6}\right)^2 - \frac{867}{36}
\]