Answer:
x = 3 + √7
x = 3 - √7
Explanation:
To solve the quadratic equation -x² + 6x - 2 = 0, we can use the quadratic formula. The quadratic formula is given by:
x = (-b ± √(b² - 4ac)) / (2a)
In this equation, a, b, and c represent the coefficients of the quadratic equation in the form ax² + bx + c = 0.
Comparing the given equation -x² + 6x - 2 = 0 with the standard form, we can determine the values of a, b, and c:
a = -1
b = 6
c = -2
Substituting these values into the quadratic formula, we have:
x = (-6 ± √(6² - 4(-1)(-2))) / (2(-1))
Simplifying further:
x = (-6 ± √(36 - 8)) / (-2)
x = (-6 ± √(28)) / (-2)
x = (-6 ± √(4 * 7)) / (-2)
x = (-6 ± 2√7) / (-2)
Now, we can simplify the expression further:
x = (6 ± 2√7) / 2
Dividing both the numerator and denominator by 2:
x = 3 ± √7
Therefore, the solutions to the quadratic equation -x² + 6x - 2 = 0 are:
x = 3 + √7
x = 3 - √7
We used the quadratic formula because it is a reliable and systematic method for solving quadratic equations. It works for any quadratic equation, regardless of whether the equation has real or imaginary solutions. By substituting the coefficients of the quadratic equation into the formula, we can determine the roots of the equation accurately.