To solve the quadratic equation [tex]\( v^2 + 10v + 25 = 0 \)[/tex], we can use the quadratic formula, which is given by:
[tex]\[
v = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are the coefficients of the quadratic equation [tex]\(av^2 + bv + c = 0\)[/tex]. In our equation:
[tex]\[
a = 1, \quad b = 10, \quad c = 25
\][/tex]
First, we identify the discriminant, which is the part under the square root in the quadratic formula:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
\Delta = 10^2 - 4 \cdot 1 \cdot 25 = 100 - 100 = 0
\][/tex]
Since the discriminant is zero, we know there is only one unique solution for [tex]\(v\)[/tex], given by:
[tex]\[
v = \frac{-b}{2a}
\][/tex]
Substituting the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[
v = \frac{-10}{2 \cdot 1} = \frac{-10}{2} = -5
\][/tex]
Thus, the quadratic equation [tex]\(v^2 + 10v + 25 = 0\)[/tex] has a double root at:
[tex]\[
v = -5
\][/tex]
Therefore, the solutions to the quadratic equation are:
[tex]\[
v = -5, \quad v = -5
\][/tex]
These values can be represented as a double root [tex]\(v = -5\)[/tex].
