find a quadratic equation with roots -1+4i and -1-4i
can someone help explain the steps?
To find a quadratic equation with given roots -1+4i and -1-4i, we can use the fact that complex roots of a quadratic equation always occur in conjugate pairs.
Step 1: Start with the standard form of a quadratic equation: ax^2 + bx + c = 0.
Step 2: Since our roots are -1+4i and -1-4i, we know that the quadratic equation will have the form (x - (-1+4i))(x - (-1-4i)) = 0, where the expressions inside the parentheses represent the factors corresponding to the complex roots.
Step 3: Simplify the equation by multiplying the factors:
(x - (-1+4i))(x - (-1-4i)) = (x + 1 - 4i)(x + 1 + 4i) = 0
Step 4: Multiply using the distributive property:
(x + 1 - 4i)(x + 1 + 4i) = (x + 1)(x + 1) - (x + 1)(4i) - (4i)(x + 1) + (4i)(-4i)
Step 5: Simplify further:
(x + 1)(x + 1) - 4i(x + 1) - 4ix - 4i + 4i^2
= (x^2 + x + x + 1) - 4ix + 4i - 4i^2
Step 6: Simplify i^2 = -1:
x^2 + 2x + 1 - 4ix + 4i + 4
= x^2 + 2x + 5 - 4ix + 4i
Step 7: Rearrange the equation into standard form:
x^2 + (2 - 4i)x + (5 + 4i) = 0
So, the quadratic equation with roots -1+4i and -1-4i is x^2 + (2 - 4i)x + (5 + 4i) = 0.