Question

Find the minimum value of the parabola y = 5x^2 − 6x − 3.

Simplify your answer and write it as a proper fraction, improper fraction, or integer.

Answer 1

To find the minimum value of the parabola given by the equation \(y = 5x^2 - 6x - 3\), we can use the formula for the vertex of a quadratic function, which occurs at \(x = -\frac{b}{2a}\).

In this equation:

- \(a = 5\)
- \(b = -6\)

Substituting these values into the vertex formula:

\[
x = -\frac{-6}{2 \cdot 5} = \frac{6}{10} = \frac{3}{5}
\]

Next, we will find the corresponding \(y\) value at \(x = \frac{3}{5}\):

\[
y = 5\left(\frac{3}{5}\right)^2 - 6\left(\frac{3}{5}\right) - 3
\]

Calculating \(\left(\frac{3}{5}\right)^2\):

\[
\left(\frac{3}{5}\right)^2 = \frac{9}{25}
\]

Now substituting back into the equation for \(y\):

\[
y = 5 \cdot \frac{9}{25} - 6 \cdot \frac{3}{5} - 3
\]

Simplifying \(5 \cdot \frac{9}{25}\):

\[
5 \cdot \frac{9}{25} = \frac{45}{25} = \frac{9}{5}
\]

Now simplifying \(6 \cdot \frac{3}{5}\):

\[
6 \cdot \frac{3}{5} = \frac{18}{5}
\]

Now we can substitute these simplified values into the \(y\) equation:

\[
y = \frac{9}{5} - \frac{18}{5} - 3
\]

Combining the fractions:

\[
y = \frac{9}{5} - \frac{18}{5} - \frac{15}{5} = \frac{9 - 18 - 15}{5} = \frac{-24}{5}
\]

Thus, the minimum value of the parabola is:

\[
\boxed{-\frac{24}{5}}
\]