To solve the given quadratic polynomial [tex]\( 2x^2 + 4x + 7 \)[/tex], we will:
1. Determine the number of roots using the fundamental theorem of algebra.
2. Find the roots using the quadratic formula.
### (a) Number of roots
According to the fundamental theorem of algebra, a polynomial of degree [tex]\( n \)[/tex] has exactly [tex]\( n \)[/tex] roots, accounting for multiplicities and including complex roots.
For the polynomial [tex]\( 2x^2 + 4x + 7 \)[/tex], which is a quadratic (degree 2), we can state that it has exactly 2 roots.
### (b) Finding the roots
To find the roots of the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex], we use the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Given the polynomial [tex]\( 2x^2 + 4x + 7 \)[/tex]:
[tex]\[
a = 2, \quad b = 4, \quad c = 7
\][/tex]
#### Step 1: Calculate the discriminant
The discriminant [tex]\(\Delta\)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
\Delta = 4^2 - 4(2)(7) = 16 - 56 = -40
\][/tex]
Since the discriminant is negative ([tex]\(\Delta = -40\)[/tex]), the roots are complex.
#### Step 2: Calculate the real and imaginary parts of the roots
The quadratic formula for complex roots is:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Since [tex]\(\Delta\)[/tex] is negative, we rewrite [tex]\(\sqrt{\Delta}\)[/tex] as [tex]\(\sqrt{-40} = i\sqrt{40}\)[/tex], where [tex]\(i\)[/tex] is the imaginary unit.
[tex]\[
x = \frac{-4 \pm i\sqrt{40}}{2 \cdot 2}
\][/tex]
[tex]\[
x = \frac{-4 \pm i\sqrt{40}}{4}
\][/tex]
[tex]\[
x = \frac{-4}{4} \pm \frac{i\sqrt{40}}{4}
\][/tex]
[tex]\[
x = -1 \pm \frac{i\sqrt{40}}{4}
\][/tex]
Simplifying further, we use [tex]\(\sqrt{40} = 2\sqrt{10}\)[/tex]:
[tex]\[
x = -1 \pm \frac{2i\sqrt{10}}{4}
\][/tex]
[tex]\[
x = -1 \pm \frac{i\sqrt{10}}{2}
\][/tex]
Thus, the roots are:
[tex]\[
x_1 = -1 + \frac{i\sqrt{10}}{2}, \quad x_2 = -1 - \frac{i\sqrt{10}}{2}
\][/tex]
Given these simplifications, we observe that the imaginary parts are:
[tex]\[
\frac{\sqrt{10}}{2} \approx 1.5811388300841898
\][/tex]
Thus, the roots can be written as:
[tex]\[
x_1 = -1 + 1.5811388300841898i, \quad x_2 = -1 - 1.5811388300841898i
\][/tex]
### Final Answer:
1. (a) The polynomial [tex]\(2x^2 + 4x + 7\)[/tex] has 2 roots.
2. (b) The roots of the polynomial [tex]\(2x^2 + 4x + 7\)[/tex] are:
[tex]\[
x_1 = -1 + 1.5811388300841898i, \quad x_2 = -1 - 1.5811388300841898i
\][/tex]
