The image you provided shows the following quadratic equation:

$2x^2 - 2x - 7 = 0$

You are asked to solve it using the quadratic formula.

The quadratic formula is:

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

Where $a$, $b$, and $c$ are the coefficients from the standard form of the quadratic equation $ax^2 + bx + c = 0$.

For your equation, we have:

• $a = 2$
• $b = -2$
• $c = -7$

Now, we substitute these values into the quadratic formula:

$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(2)(-7)}}{2(2)}$ $x = \frac{2 \pm \sqrt{4 + 56}}{4}$ $x = \frac{2 \pm \sqrt{60}}{4}$

Simplifying the square root of 60: $\sqrt{60} = \sqrt{4 \times 15} = 2\sqrt{15}$

Now substitute: $x = \frac{2 \pm 2\sqrt{15}}{4}$

We can simplify further: $x = \frac{2}{4} \pm \frac{2\sqrt{15}}{4}$ $x = \frac{1}{2} \pm \frac{\sqrt{15}}{2}$

Thus, the two possible solutions are: $x = \frac{1 + \sqrt{15}}{2} \quad \text{or} \quad x = \frac{1 - \sqrt{15}}{2}$

These are the solutions for the given quadratic equation.

Would you like further details on any steps or have any questions? Here are some related questions to expand your learning:

1. What happens when the discriminant $b^2 - 4ac$ is negative in the quadratic formula?
2. How can you simplify square roots of non-perfect squares in the quadratic formula?
3. What is the geometric interpretation of the quadratic formula solutions on a graph?
4. How would you solve a quadratic equation by factoring instead of using the quadratic formula?
5. What are some practical applications of the quadratic formula in real life?

Tip: Always check if factoring is an easier method before applying the quadratic formula!