Question

solve for (x) in the equation (x^2 - 12x + 59 = 0).

• (x = -12 pm sqrt{85})
• (x = -6 pm sqrt{23}i)
• (x = 6 pm sqrt{23}i)
• (x = 12 pm sqrt{85})

Explanation:

Step1: Identify the coefficients

$$a = 1, \quad b = -12, \quad c = 59$$

Step2: Apply the quadratic formula

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Step3: Substitute the values

$$x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(1)(59)}}{2(1)}$$

Step4: Simplify the expression

$$x = \frac{12 \pm \sqrt{144 - 236}}{2}$$

Step5: Simplify the radicand

$$x = \frac{12 \pm \sqrt{-92}}{2}$$

Step6: Express with imaginary unit

$$x = \frac{12 \pm 2\sqrt{23}i}{2}$$

Step7: Divide by the denominator

$$x = 6 \pm \sqrt{23}i$$

Answer:

\(x = 6 \pm \sqrt{23}i\)