Find the zeros of the following quadratic using the quadratic formula and write down all your work in your written journal.𝑓(𝑥)=2𝑥2+2𝑥−5

1. Break Down the Problem

We need to find the zeros of the quadratic function given by:

$f(x) = 2x^2 + 2x - 5$

To find the zeros, we can use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a = 2$, $b = 2$, and $c = -5$.

2. Relevant Concepts

The quadratic formula will help us calculate the values for $x$ where $f(x) = 0$.

3. Analysis and Detail

1. Identify coefficients:

   • $a = 2$
   • $b = 2$
   • $c = -5$

2. Calculate the discriminant: $D = b^2 - 4ac = 2^2 - 4 \cdot 2 \cdot (-5)$ $D = 4 + 40 = 44$

3. Use the quadratic formula to find the zeros: $x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-2 \pm \sqrt{44}}{2 \cdot 2}$ Simplifying $\sqrt{44}$: $\sqrt{44} = \sqrt{4 \cdot 11} = 2\sqrt{11}$

4. Substitute back into the formula: $x = \frac{-2 \pm 2\sqrt{11}}{4}$ Simplifying further: $x = \frac{-1 \pm \sqrt{11}}{2}$

4. Verify and Summarize

The zeros of the quadratic function are: $x_1 = \frac{-1 + \sqrt{11}}{2} \quad \text{and} \quad x_2 = \frac{-1 - \sqrt{11}}{2}$

Final Answer

The zeros of the quadratic function $f(x) = 2x^2 + 2x - 5$ are: $x_1 = \frac{-1 + \sqrt{11}}{2}, \quad x_2 = \frac{-1 - \sqrt{11}}{2}$