use the quadratic formula with a=6, b=-8 and c=-1

To solve the quadratic equation \(6x^2 - 8x - 1 = 0\) using the quadratic formula, we follow these steps:

The quadratic formula is:

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

Given:
- \(a = 6\)
- \(b = -8\)
- \(c = -1\)

### Step 1: Calculate the discriminant
\[
\text{Discriminant} = b^2 - 4ac = (-8)^2 - 4(6)(-1) = 64 + 24 = 88
\]

### Step 2: Apply the quadratic formula
\[
x = \frac{-(-8) \pm \sqrt{88}}{2(6)} = \frac{8 \pm \sqrt{88}}{12}
\]

### Step 3: Simplify the square root (if possible)
\[
\sqrt{88} = \sqrt{4 \times 22} = 2\sqrt{22}
\]

So the equation becomes:
\[
x = \frac{8 \pm 2\sqrt{22}}{12}
\]

### Step 4: Simplify the expression
\[
x = \frac{8}{12} \pm \frac{2\sqrt{22}}{12} = \frac{2}{3} \pm \frac{\sqrt{22}}{6}
\]

Thus, the two solutions are:
\[
x_1 = \frac{2}{3} + \frac{\sqrt{22}}{6}
\]
\[
x_2 = \frac{2}{3} - \frac{\sqrt{22}}{6}
\]

These are the exact solutions for \(x\).

Would you like to know the decimal approximations or further details on any step?

---

**Relative Questions:**
1. What does the discriminant tell us about the nature of the roots?
2. How would the solution change if the discriminant were negative?
3. Can the quadratic formula be used if \(a = 0\)? Why or why not?
4. How does the quadratic formula relate to completing the square?
5. What are some practical applications of quadratic equations in real life?

**Tip:** When using the quadratic formula, always check the discriminant first to determine the nature of the roots (real and distinct, real and repeated, or complex).

Learn how to solve the quadratic equation 6x^2 - 8x - 1 = 0 using the quadratic formula step-by-step. Understand the discriminant and how it determines the nature of the roots. Suitable for high school students studying algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation