QUESTION 15 Find the coordinates for the vertex of the following quadratic function algebraically. Be prepared to show your algebraic work. $$ \mathbf{P}(\mathbf{x})=2 \mathrm{x}^{2}-5 \mathbf{x} $$ Enter the exact coordinates separately in each blank. (Do not round values.) The vertex is (

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To find the coordinates of the vertex of the quadratic function $$ P(x) = 2x^2 - 5x $$, we can use the vertex formula. The x-coordinate of the vertex for a quadratic function in the form $$ ax^2 + bx + c $$ is given by:

$$
x = -\frac{b}{2a}
$$

Here, $$ a = 2 $$ and $$ b = -5 $$.

### Step 1: Calculate the x-coordinate of the vertex

Substituting the values of $$ a $$ and $$ b $$ into the formula:

$$
x = -\frac{-5}{2 \cdot 2} = \frac{5}{4}
$$

### Step 2: Calculate the y-coordinate of the vertex

Now, we need to find the y-coordinate by substituting $$ x = \frac{5}{4} $$ back into the function $$ P(x) $$:

$$
P\left(\frac{5}{4}\right) = 2\left(\frac{5}{4}\right)^2 - 5\left(\frac{5}{4}\right)
$$

Calculating $$ \left(\frac{5}{4}\right)^2 $$:

$$
\left(\frac{5}{4}\right)^2 = \frac{25}{16}
$$

Now substituting this back into the equation:

$$
P\left(\frac{5}{4}\right) = 2 \cdot \frac{25}{16} - 5 \cdot \frac{5}{4}
$$

Calculating $$ 2 \cdot \frac{25}{16} $$:

$$
2 \cdot \frac{25}{16} = \frac{50}{16} = \frac{25}{8}
$$

Calculating $$ 5 \cdot \frac{5}{4} $$:

$$
5 \cdot \frac{5}{4} = \frac{25}{4}
$$

Now we need a common denominator to combine these two fractions. The common denominator between 8 and 4 is 8:

$$
\frac{25}{4} = \frac{50}{8}
$$

Now substituting back into the equation:

$$
P\left(\frac{5}{4}\right) = \frac{25}{8} - \frac{50}{8} = -\frac{25}{8}
$$

### Final Result

The coordinates of the vertex are:

$$
\left( \frac{5}{4}, -\frac{25}{8} \right)
$$

Thus, the vertex is:

- x-coordinate: $$ \frac{5}{4} $$
- y-coordinate: $$ -\frac{25}{8} $$

So, the final answer is:

The vertex is $$ \left( \frac{5}{4}, -\frac{25}{8} \right) $$.
The vertex is $$ \left( \frac{5}{4}, -\frac{25}{8} \right) $$.

QUESTION 15 Find the coordinates for the vertex of the following quadratic function algebraically. Be prepared to show your algebraic work. \( \mathbf{P}(\mathbf{x})=2 \mathrm{x}^{2}-5 \mathbf{x} \) Enter the exact coordinates separately in each blank. (Do not round values.) The vertex is (

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