Questions: Use the quadratic formula to solve the equation (x^2-8x+65=0). Enter multiple answers as a list separated by commas. Example: (2+2i, 2-2i)

Solution Steps

To solve the quadratic equation \(x^2 - 8x + 65 = 0\) using the quadratic formula, we need to identify the coefficients \(a\), \(b\), and \(c\) from the equation \(ax^2 + bx + c = 0\). Then, we apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the roots.

1. Identify the coefficients \(a\), \(b\), and \(c\) from the equation.
2. Compute the discriminant \(D = b^2 - 4ac\).
3. Use the quadratic formula to find the roots.

The given quadratic equation is \(x^2 - 8x + 65 = 0\). From this equation, we identify the coefficients:

• \(a = 1\)
• \(b = -8\)
• \(c = 65\)

We calculate the discriminant \(D\) using the formula: \[ D = b^2 - 4ac \] Substituting the values: \[ D = (-8)^2 - 4 \cdot 1 \cdot 65 = 64 - 260 = -196 \]

Since the discriminant \(D\) is negative, the roots will be complex. We use the quadratic formula: \[ x = \frac{-b \pm \sqrt{D}}{2a} \] Substituting the values: \[ x = \frac{-(-8) \pm \sqrt{-196}}{2 \cdot 1} = \frac{8 \pm \sqrt{196}i}{2} \] Calculating further: \[ x = \frac{8 \pm 14i}{2} = 4 \pm 7i \]

Final Answer