Solve the equation 2x^2 - 8x +7 = 0. Give your answers in surd form.

To solve the quadratic equation \(2x^2 - 8x + 7 = 0\), we'll use the quadratic formula:

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

For the equation \(2x^2 - 8x + 7 = 0\), the coefficients are:
- \(a = 2\)
- \(b = -8\)
- \(c = 7\)

### Step 1: Calculate the discriminant
The discriminant \(\Delta\) is given by:

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

Substitute the values of \(a\), \(b\), and \(c\):

\[
\Delta = (-8)^2 - 4(2)(7) = 64 - 56 = 8
\]

### Step 2: Apply the quadratic formula
Now that we have the discriminant, we can substitute everything into the quadratic formula:

\[
x = \frac{-(-8) \pm \sqrt{8}}{2(2)}
\]

Simplify the expression:

\[
x = \frac{8 \pm \sqrt{8}}{4}
\]

### Step 3: Simplify the surd
The square root of 8 can be simplified:

\[
\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}
\]

So, the expression for \(x\) becomes:

\[
x = \frac{8 \pm 2\sqrt{2}}{4}
\]

Simplify further by dividing each term in the numerator by 4:

\[
x = \frac{8}{4} \pm \frac{2\sqrt{2}}{4}
\]

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

### Final Answer
The solutions to the equation \(2x^2 - 8x + 7 = 0\) in surd form are:

\[
x = 2 + \frac{\sqrt{2}}{2} \quad \text{or} \quad x = 2 - \frac{\sqrt{2}}{2}
\]

Would you like any details or have any questions?

Here are some related questions:
1. How do you solve a quadratic equation using the quadratic formula?
2. What is the discriminant, and how does it determine the nature of the roots?
3. How can you simplify square roots (surds)?
4. What happens if the discriminant is zero or negative?
5. How do you complete the square for a quadratic equation?

**Tip:** Always simplify surds to their simplest form for cleaner and more accurate results.

Learn how to solve the quadratic equation 2x^2 - 8x + 7 in surd form with a detailed step-by-step solution. This problem covers key algebraic concepts, including the quadratic formula and simplifying surds. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation