Certainly! Let's solve the quadratic equation [tex]\( x^2 - 2x + 30 = 0 \)[/tex] step by step.
The general form of a quadratic equation is [tex]\( ax^2 + bx + c = 0 \)[/tex]. Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = 30 \)[/tex].
To find the roots of the quadratic equation, we use the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
First, let's calculate the discriminant, which is the expression under the square root in the quadratic formula:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substitute [tex]\( a = 1 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = 30 \)[/tex] into the discriminant formula:
[tex]\[
\Delta = (-2)^2 - 4 \cdot 1 \cdot 30
\][/tex]
[tex]\[
\Delta = 4 - 120
\][/tex]
[tex]\[
\Delta = -116
\][/tex]
The discriminant, [tex]\(\Delta\)[/tex], is [tex]\(-116\)[/tex]. Since the discriminant is less than zero, we conclude that:
1. The discriminant is negative, which means the equation has no real roots.
Thus, the solution to the quadratic equation [tex]\( x^2 - 2x + 30 = 0 \)[/tex] is that it has no real roots.
