Question

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

Answer 1

To determine the number of real solutions for the equation \( x^2 + 10x = -21 \), we first rearrange it into standard form. This gives us:

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

Now, we can identify the coefficients \( a \), \( b \), and \( c \) from the standard quadratic form \( ax^2 + bx + c = 0 \):

- \( a = 1 \)
- \( b = 10 \)
- \( c = 21 \)

Next, we will calculate the discriminant, which is given by the formula:

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

Substituting the values of \( a \), \( b \), and \( c \) into the discriminant formula:

\[
D = (10)^2 - 4 \cdot 1 \cdot 21
\]

Calculating the terms:

\[
D = 100 - 84 = 16
\]

Now we analyze the value of the discriminant:

- If \( D > 0 \), there are two distinct real solutions.
- If \( D = 0 \), there is one real solution (a repeated root).
- If \( D < 0 \), there are no real solutions.

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