find a quadratic equation with roots -1+4i and -1-4i
If someone could explain it would help me understand
To find a quadratic equation with the given roots, we can use the fact that the sum of the roots of a quadratic equation is equal to the negation of the coefficient of the linear term, and the product of the roots is equal to the constant term divided by the coefficient of the quadratic term.
Let's break down the steps:
Step 1: Identify the Form of the Quadratic Equation
A quadratic equation is typically written in the form of ax^2 + bx + c = 0, where a, b, and c are constants.
Step 2: Utilize the Given Roots
The given roots are -1+4i and -1-4i. Since complex roots always appear in conjugate pairs (i.e., if a+bi is a root, then a-bi is also a root), we know that -1+4i and -1-4i are both roots of the quadratic equation.
Step 3: Find the Coefficients
To find the quadratic equation, we need to find the values of a, b, and c.
Step 4: Use the Sum and Product of Roots Formulas
The sum of the roots is equal to the negation of the coefficient of the linear term (b), i.e., -b.
The product of the roots is equal to the constant term (c) divided by the coefficient of the quadratic term (a), i.e., c/a.
Using the sum and product formulas, we can write the following equations:
-1+4i + -1-4i = -b/ a
(-1+4i)(-1-4i) = c/ a
Step 5: Simplify the Equations
Let's simplify these equations:
-2 = -b/ a
(1-16i^2) = c/ a [Using the fact that i^2 = -1]
Simplified, these equations become:
-2 = -b/ a
(1-16(-1)) = c/ a
-2 = -b/ a
(1+16) = c/ a
-2 = -b/ a
17 = c/ a
From the first equation, we get b = 2a.
Step 6: Write the Quadratic Equation
Now that we have values for b and c in terms of a, we can substitute them back into the original quadratic equation form.
a(x^2) + b(x) + c = 0
Substituting b = 2a and c = 17a, we get:
a(x^2) + 2a(x) + 17a = 0
Factoring out 'a', we obtain the quadratic equation:
ax^2 + 2ax + 17a = 0
To make this equation unique (since a can be any non-zero constant), we can divide the entire equation by 'a':
x^2 + 2x + 17 = 0
This is the quadratic equation we were looking for, with roots -1+4i and -1-4i.