Question
Solve the equation for all real solutions in simplest form. \[ x^{2}-7 x+2=0 \]
Ask by Medina Todd. in the United States
Jan 22,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The solutions are \( x = \frac{7 + \sqrt{41}}{2} \) and \( x = \frac{7 - \sqrt{41}}{2} \).
Solution
To solve the quadratic equation:
\[
x^{2} - 7x + 2 = 0
\]
we can use the **quadratic formula**, which is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For the equation \( ax^2 + bx + c = 0 \):
- \( a = 1 \)
- \( b = -7 \)
- \( c = 2 \)
Plugging these values into the quadratic formula:
\[
x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(1)(2)}}{2(1)} = \frac{7 \pm \sqrt{49 - 8}}{2} = \frac{7 \pm \sqrt{41}}{2}
\]
So, the solutions are:
\[
x = \frac{7 + \sqrt{41}}{2} \quad \text{and} \quad x = \frac{7 - \sqrt{41}}{2}
\]
**Final Answer:**
All real solutions are (7 ± √41) divided by two. In symbols,
x = ( 7 ± √41 ) ⁄ 2
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
To solve the quadratic equation \( x^{2}-7 x+2=0 \), we can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \( a = 1 \), \( b = -7 \), and \( c = 2 \). First, calculate the discriminant: \[ b^2 - 4ac = (-7)^2 - 4(1)(2) = 49 - 8 = 41 \] Now substitute into the quadratic formula: \[ x = \frac{7 \pm \sqrt{41}}{2} \] Thus, the solutions in simplest form are: \[ x = \frac{7 + \sqrt{41}}{2} \quad \text{and} \quad x = \frac{7 - \sqrt{41}}{2} \]
