Question
\( x^{2}-8 x+20=0 \) Write one exact, simplified solution in each box. You can add or remove boxes. If a solution is not a real number, write it in the form a + bi or \( \mathrm{a}-\mathrm{bi} \), where a and b are real numbers.
Ask by Johnson Gibson. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The solutions are \( 4 + 2i \) and \( 4 - 2i \).
Solution
The solutions to the equation \( x^{2} - 8x + 20 = 0 \) are complex numbers. Using the quadratic formula:
\[
x = \frac{8 \pm \sqrt{64 - 80}}{2} = \frac{8 \pm \sqrt{-16}}{2} = \frac{8 \pm 4i}{2} = 4 \pm 2i
\]
**Solutions:**
1. \( 4 + 2i \)
2. \( 4 - 2i \)
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Simplify this solution Bonus Knowledge
To solve the quadratic equation \( x^2 - 8x + 20 = 0 \), we can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \( a = 1 \), \( b = -8 \), and \( c = 20 \). Calculating the discriminant: \[ b^2 - 4ac = (-8)^2 - 4(1)(20) = 64 - 80 = -16 \] Since the discriminant is negative, the solutions will be complex. Now applying the quadratic formula: \[ x = \frac{8 \pm \sqrt{-16}}{2 \cdot 1} = \frac{8 \pm 4i}{2} = 4 \pm 2i \] The two solutions are: 1. \( 4 + 2i \) 2. \( 4 - 2i \)
