Question

Use the discriminant to determine the number of real solutions of x2+10x=−21
.(1 point)

Answer 1

To determine the number of real solutions for the equation \(x^2 + 10x = -21\), we first need to rearrange it into standard form \(ax^2 + bx + c = 0\).

Starting with the given equation:

\[
x^2 + 10x + 21 = 0
\]

Now we identify the coefficients:
- \(a = 1\)
- \(b = 10\)
- \(c = 21\)

Next, we calculate the discriminant \(D\) using the formula:

\[
D = b^2 - 4ac
\]

Substituting in the values of \(a\), \(b\), and \(c\):

\[
D = (10)^2 - 4(1)(21)
\]
\[
D = 100 - 84
\]
\[
D = 16
\]

Now, we evaluate the discriminant:
- If \(D > 0\): there are 2 distinct real solutions.
- If \(D = 0\): there is exactly 1 real solution (a repeated solution).
- If \(D < 0\): there are no real solutions.

Since \(D = 16\) which is greater than 0, we conclude that there are **2 distinct real solutions** to the equation \(x^2 + 10x = -21\).