Question
Solve the quadratic equ -8 \( x^{2}-6 x-2=0 \)
Ask by Summers Bolton. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The solutions to the equation \( x^{2} - 6x - 2 = 0 \) are \( x = 3 + \sqrt{11} \) and \( x = 3 - \sqrt{11} \).
Solution
To solve the quadratic equation:
\[
x^{2} - 6x - 2 = 0
\]
we can use the **quadratic formula**, which is:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For the given equation, the coefficients are:
- \( a = 1 \)
- \( b = -6 \)
- \( c = -2 \)
**Step 1: Calculate the Discriminant (\( \Delta \))**
\[
\Delta = b^2 - 4ac = (-6)^2 - 4(1)(-2) = 36 + 8 = 44
\]
**Step 2: Apply the Quadratic Formula**
\[
x = \frac{-(-6) \pm \sqrt{44}}{2 \times 1} = \frac{6 \pm \sqrt{44}}{2}
\]
Simplify \( \sqrt{44} \):
\[
\sqrt{44} = \sqrt{4 \times 11} = 2\sqrt{11}
\]
So,
\[
x = \frac{6 \pm 2\sqrt{11}}{2} = 3 \pm \sqrt{11}
\]
**Solutions:**
\[
x = 3 + \sqrt{11} \quad \text{and} \quad x = 3 - \sqrt{11}
\]
**Approximate Values:**
- \( \sqrt{11} \approx 3.3166 \)
- \( x \approx 3 + 3.3166 = 6.3166 \)
- \( x \approx 3 - 3.3166 = -0.3166 \)
**Final Answer:**
\[
x = 3 + \sqrt{11} \quad \text{and} \quad x = 3 - \sqrt{11}
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
To solve the quadratic equation \(x^{2}-6x-2=0\), we can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] In this equation, \(a = 1\), \(b = -6\), and \(c = -2\). Plugging these values into the formula gives us: \[ x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot (-2)}}{2 \cdot 1} \] Calculating this step-by-step, we first find: \[ b^2 - 4ac = 36 + 8 = 44 \] So the formula becomes: \[ x = \frac{6 \pm \sqrt{44}}{2} \] Now, simplify \(\sqrt{44}\) to \(2\sqrt{11}\): \[ x = \frac{6 \pm 2\sqrt{11}}{2} \] Dividing everything by 2, we get: \[ x = 3 \pm \sqrt{11} \] Thus, the solutions are: \[ x = 3 + \sqrt{11} \quad \text{and} \quad x = 3 - \sqrt{11} \]
