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