Question
Solve the quadratic equation. \[ x^{2}-10 x-4=0 \] Write one exact, simplified solution in each box. You can add or remove boxes. If a solutio not a real number, write it in the form a + bi or a - bi, where a and b are real numbers. \( \frac{\square}{\square} \),
Ask by Osborne Wright. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The solutions are \( 5 + \sqrt{29} \) and \( 5 - \sqrt{29} \).
Solution
To solve the quadratic equation:
\[
x^{2} - 10x - 4 = 0
\]
We can use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \( a = 1 \), \( b = -10 \), and \( c = -4 \).
Plugging in the values:
\[
x = \frac{10 \pm \sqrt{(-10)^2 - 4(1)(-4)}}{2(1)} = \frac{10 \pm \sqrt{100 + 16}}{2} = \frac{10 \pm \sqrt{116}}{2}
\]
Simplify the square root:
\[
\sqrt{116} = \sqrt{4 \times 29} = 2\sqrt{29}
\]
So the solutions become:
\[
x = \frac{10 \pm 2\sqrt{29}}{2} = 5 \pm \sqrt{29}
\]
**Solutions:**
\[
5 + \sqrt{29}, \quad 5 - \sqrt{29}
\]
Reviewed and approved by the UpStudy tutoring team
Like
error msg
Explain 
Simplify this solution Beyond the Answer
To solve the quadratic equation \( x^{2}-10 x-4=0 \), we can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \( a = 1 \), \( b = -10 \), and \( c = -4 \). First, we calculate the discriminant: \[ b^2 - 4ac = (-10)^2 - 4(1)(-4) = 100 + 16 = 116 \] Now, applying the quadratic formula: \[ x = \frac{10 \pm \sqrt{116}}{2} \] Simplifying \( \sqrt{116} \): \[ \sqrt{116} = \sqrt{4 \times 29} = 2\sqrt{29} \] So we have: \[ x = \frac{10 \pm 2\sqrt{29}}{2} = 5 \pm \sqrt{29} \] The two exact, simplified solutions are: \( 5 + \sqrt{29} \), \( 5 - \sqrt{29} \)
