Find all solutions of the equation and express them in the form a + bi.x^2 − 16x + 65 = 0
To find the solutions of the given equation, we can use the quadratic formula. The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are coefficients.
In this case, the equation is in the form x^2 − 16x + 65 = 0, which means a = 1, b = -16, and c = 65.
The quadratic formula states that the solutions of a quadratic equation ax^2 + bx + c = 0 can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Let's substitute the values a = 1, b = -16, and c = 65 into the quadratic formula:
x = (16 ± √((-16)^2 - 4(1)(65))) / (2(1))
Simplifying further:
x = (16 ± √(256 - 260)) / 2
x = (16 ± √(-4)) / 2
At this point, we have a square root of a negative number, which means the solutions will involve complex numbers. Let's simplify the expression further:
x = (16 ± 2i) / 2
Dividing both numerator and denominator by 2:
x = 8 ± i
Therefore, the solutions of the equation x^2 − 16x + 65 = 0 in the form a + bi are:
x = 8 + i
x = 8 - i