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

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 \( x^2 + 10x + 21 = 0 \).