Question

what are the roots of the equation 25x^2+40x+17=0 in simplest a+bi form

Answer 1

To find the roots of the quadratic equation \( 25x^2 + 40x + 17 = 0 \), we can use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

where \( a = 25 \), \( b = 40 \), and \( c = 17 \).

First, we calculate the discriminant \( b^2 - 4ac \):

\[
b^2 = 40^2 = 1600
\]
\[
4ac = 4 \cdot 25 \cdot 17 = 1700
\]
\[
b^2 - 4ac = 1600 - 1700 = -100
\]

Since the discriminant is negative (\(-100\)), this means the roots are complex.

Now we can substitute these values into the quadratic formula:

\[
x = \frac{-40 \pm \sqrt{-100}}{2 \cdot 25}
\]
\[
x = \frac{-40 \pm 10i}{50}
\]

Now, we simplify:

\[
x = \frac{-40}{50} \pm \frac{10i}{50}
\]
\[
x = -\frac{4}{5} \pm \frac{1}{5}i
\]

Thus, the roots of the equation \( 25x^2 + 40x + 17 = 0 \) in simplest \( a + bi \) form are:

\[
x = -\frac{4}{5} + \frac{1}{5}i \quad \text{and} \quad x = -\frac{4}{5} - \frac{1}{5}i
\]