Question
Solve the quadratic equation. \[ x^{2}-x-10=0 \] Write one exact solution in each box. You can add or remove boxes. If there are no real solutions, remove all boxes.
Ask by Dunn Alexander. in the United States
Feb 26,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The solutions are:
- Box 1: \( x = \frac{1 + \sqrt{41}}{2} \)
- Box 2: \( x = \frac{1 - \sqrt{41}}{2} \)
Solution
The quadratic equation is
x² - x - 10 = 0.
We use the quadratic formula:
x = ( -b ± √(b² - 4ac) ) / (2a),
where a = 1, b = -1, and c = -10.
First, compute the discriminant:
b² - 4ac = (-1)² - 4(1)(-10) = 1 + 40 = 41.
Since the discriminant is positive, there are two real solutions given by:
x = (1 ± √41) / 2.
One exact solution in each box:
Box 1: x = (1 + √41) / 2
Box 2: x = (1 - √41) / 2
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Simplify this solution Mind Expander
To solve the quadratic equation \( x^{2}-x-10=0 \), we can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \( a = 1 \), \( b = -1 \), and \( c = -10 \). Calculating the discriminant: \[ b^2 - 4ac = (-1)^2 - 4(1)(-10) = 1 + 40 = 41 \] Now we plug the values into the quadratic formula: \[ x = \frac{-(-1) \pm \sqrt{41}}{2(1)} = \frac{1 \pm \sqrt{41}}{2} \] So the solutions are: \[ x_1 = \frac{1 + \sqrt{41}}{2} \] \[ x_2 = \frac{1 - \sqrt{41}}{2} \] Here are the exact solutions: **Box 1:** \( \frac{1 + \sqrt{41}}{2} \) **Box 2:** \( \frac{1 - \sqrt{41}}{2} \)
