Using the discriminant, answer the quadratic equation : 2x^2-2(square root with 10x inside)+5

The quadratic equation you provided is:

\[
2x^2 - 2\sqrt{10}x + 5 = 0
\]

To solve this using the discriminant, let's follow these steps:

### Step 1: Identify the coefficients
For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant \( \Delta \) is given by the formula:

\[
\Delta = b^2 - 4ac
\]

Here, comparing your equation with the standard form:
- \( a = 2 \)
- \( b = -2\sqrt{10} \)
- \( c = 5 \)

### Step 2: Calculate the discriminant

\[
\Delta = (-2\sqrt{10})^2 - 4(2)(5)
\]

First, calculate \( (-2\sqrt{10})^2 \):

\[
(-2\sqrt{10})^2 = 4 \times 10 = 40
\]

Now, calculate \( 4ac \):

\[
4 \times 2 \times 5 = 40
\]

Thus, the discriminant is:

\[
\Delta = 40 - 40 = 0
\]

### Step 3: Interpret the discriminant
When \( \Delta = 0 \), the quadratic equation has **exactly one real solution** (a repeated root). The solution is given by:

\[
x = \frac{-b}{2a}
\]

### Step 4: Solve for \( x \)

Now, substitute the values of \( b \) and \( a \):

\[
x = \frac{-(-2\sqrt{10})}{2 \times 2} = \frac{2\sqrt{10}}{4} = \frac{\sqrt{10}}{2}
\]

### Final Answer:
The solution to the quadratic equation \( 2x^2 - 2\sqrt{10}x + 5 = 0 \) is:

\[
x = \frac{\sqrt{10}}{2}
\]

Would you like more details on any part of the solution?

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Here are 5 related questions to expand your understanding:

1. How does the discriminant help in determining the nature of roots for quadratic equations?
2. What happens if the discriminant is positive or negative?
3. Can you solve another quadratic equation where the discriminant is negative?
4. How do you factor quadratic equations when the discriminant is zero?
5. What is the significance of the coefficient of \(x^2\) in determining the parabola's direction?

**Tip**: Always check the discriminant first—it can save you time by telling you whether you have real or complex solutions!

Using the discriminant, answer the quadratic equation: 2x^2 - 2(square root with 10x inside) + 5

Learn how to solve the quadratic equation 2x^2 - 2√10x + 5 using the discriminant method. This step-by-step solution covers important algebraic concepts and formulas, including calculating the discriminant and solving for roots. Perfect for students in Grades 9-10.

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