Q. Use the quadratic formula to solve the equation. \(x^2 + 10x + 25 = 0\).

Solution using the quadratic formula

1. Identify the coefficients.For the quadratic equation \(x^{2} + 10x + 25 = 0\), the coefficients are

   \(a = 1\), \(b = 10\), \(c = 25\).

2. Write the quadratic formula.The quadratic formula that gives the solutions of \(ax^{2} + bx + c = 0\) is

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

3. Compute the discriminant.The discriminant is \(b^{2} – 4ac\).
   Substitute the values of \(a\), \(b\), and \(c\):
   \[
   b^{2} – 4ac = 10^{2} – 4(1)(25).
   \]
   Evaluate each part:
   \[
   10^{2} = 100,\qquad 4(1)(25) = 100,
   \]
   so
   \[
   b^{2} – 4ac = 100 – 100 = 0.
   \]
   A discriminant of \(0\) means there is exactly one real solution (a repeated root).
4. Substitute into the quadratic formula and simplify.Substitute \(b^{2} – 4ac = 0\), \(b = 10\), and \(a = 1\) into the formula:
   \[
   x = \frac{-10 \pm \sqrt{0}}{2(1)}.
   \]
   Since \(\sqrt{0} = 0\), this becomes
   \[
   x = \frac{-10 \pm 0}{2} = \frac{-10}{2}.
   \]
   Evaluate the fraction:
   \[
   x = -5.
   \]
5. State the final solution and its multiplicity.The quadratic equation \(x^{2} + 10x + 25 = 0\) has a single (repeated) real root:
   \[
   x = -5,
   \]
   which is a root of multiplicity 2 because the discriminant is zero and the quadratic factors as \((x + 5)^{2} = 0\).