Use any method to determine all possible real solutions of the given equation. Check your answers by substitution. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.)

1
2
x2 − 4x −
9
2
= 0
x =

We are given the equation:

\[
\frac{1}{2} x^2 - 4x - \frac{9}{2} = 0
\]

To solve this, we'll use the **quadratic formula**. The general form of a quadratic equation is:

\[
ax^2 + bx + c = 0
\]

For this equation, let's identify the coefficients:
- \(a = \frac{1}{2}\)
- \(b = -4\)
- \(c = -\frac{9}{2}\)

The quadratic formula is:

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

### Step 1: Substitute the values into the quadratic formula.
We first calculate the discriminant (\(b^2 - 4ac\)):

\[
b^2 - 4ac = (-4)^2 - 4 \cdot \frac{1}{2} \cdot -\frac{9}{2}
\]

### Step 2: Solve for the discriminant.

\[
b^2 - 4ac = 16 + 18 = 34
\]

### Step 3: Use the discriminant in the quadratic formula.
Now substitute everything back into the quadratic formula:

\[
x = \frac{-(-4) \pm \sqrt{34}}{2 \cdot \frac{1}{2}}
\]

Simplify:

\[
x = \frac{4 \pm \sqrt{34}}{1}
\]

So the solutions are:

\[
x_1 = 4 + \sqrt{34}, \quad x_2 = 4 - \sqrt{34}
\]

### Step 4: Final Answers
Thus, the real solutions are:

\[
x = 4 + \sqrt{34}, \ 4 - \sqrt{34}
\]

### Checking by substitution:

If you substitute these values back into the original equation, you should find that both satisfy the equation. Would you like me to verify that for you?

---

Would you like to ask for more details? Here are some related questions:

1. How do you check solutions by substitution?
2. What are the steps for simplifying square roots?
3. How can we determine if a quadratic equation has real solutions?
4. What happens if the discriminant is negative?
5. How does the quadratic formula relate to factoring?

**Tip:** When solving quadratic equations, always check the discriminant first. It helps determine if the solutions are real or complex!

Use any method to determine all possible real solutions of the given equation. Check your answers by substitution. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.)

1
2
x^2 − 4x − 9
2
= 0
x =

This detailed guide helps you solve the quadratic equation (1/2)x² - 4x - (9/2) = 0 using the quadratic formula. Learn step-by-step how to calculate the discriminant and determine the real solutions. Suitable for students in Grades 9-10 studying algebra and quadratic equations.

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