Question

Find the roots of the equation below.

x² - 6x + 12 = 0
O-3± √3
O 3 ± √3
O-3± N3
3 ± N√3

Answer 1

We can use the quadratic formula to find the roots of the equation x² - 6x + 12 = 0.

The quadratic formula states that for an equation of the form ax² + bx + c = 0, the roots are given by:

x = (-b ± sqrt(b² - 4ac)) / 2a

Comparing the given equation to the standard form ax² + bx + c = 0, we have a = 1, b = -6, and c = 12. Substituting these values into the quadratic formula, we get:

x = (-(-6) ± sqrt((-6)² - 4(1)(12))) / 2(1)

Simplifying:

x = (6 ± sqrt(6² - 48)) / 2

x = (6 ± sqrt(-12)) / 2

The expression sqrt(-12) involves an imaginary number, so the roots of the equation are complex conjugates:

x = 3 + sqrt(3)i and x = 3 - sqrt(3)i

Therefore, the roots of the equation x² - 6x + 12 = 0 are x = 3 + sqrt(3)i and x = 3 - sqrt(3)i.

Answer 2

To find the roots of the equation x² - 6x + 12 = 0, we can use the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation.

In this case, a = 1, b = -6, and c = 12. Substituting these values into the formula, we get:

x = (6 ± √(6² - 4(1)(12))) / 2(1)
x = (6 ± √(36 - 48)) / 2
x = (6 ± √(-12)) / 2

Since the square root of a negative number is not a real number, the roots of the equation are complex numbers. We can simplify the expression for the roots by factoring out -1 from the square root:

x = (6 ± √(4 × -3)) / 2
x = (6 ± 2√(-3)) / 2
x = 3 ± √(-3)

Using the imaginary unit i, which is defined as i² = -1, we can write the roots in the form a + bi:

x = 3 ± √3i

Therefore, the roots of the equation x² - 6x + 12 = 0 are 3 + √3i and 3 - √3i.