To solve the quadratic equation [tex]\( x^2 + 10x + 21 = 0 \)[/tex], we can use the factorization method.
Step-by-Step Solution:
1. Write down the quadratic equation:
[tex]\[
x^2 + 10x + 21 = 0
\][/tex]
2. Factorize the quadratic expression:
To factor the quadratic expression, we look for two numbers that multiply to the constant term (21) and add up to the linear coefficient (10).
3. Find the factors of 21 that add up to 10:
- The factors of 21 are (1, 21), (3, 7), and their respective negative pairs.
- We need the pair that adds up to 10:
[tex]\[
3 + 7 = 10
\][/tex]
Therefore, the numbers are 3 and 7.
4. Write the quadratic expression as a product of binomials:
[tex]\[
x^2 + 10x + 21 = (x + 3)(x + 7)
\][/tex]
5. Set each factor equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
(x + 3)(x + 7) = 0
\][/tex]
This gives us two equations:
[tex]\[
x + 3 = 0 \quad \text{or} \quad x + 7 = 0
\][/tex]
6. Solve each equation for [tex]\( x \)[/tex]:
[tex]\[
x + 3 = 0 \implies x = -3
\][/tex]
[tex]\[
x + 7 = 0 \implies x = -7
\][/tex]
So, the solutions to the quadratic equation [tex]\( x^2 + 10x + 21 = 0 \)[/tex] are [tex]\( x = -3 \)[/tex] and [tex]\( x = -7 \)[/tex].
Answer:
A. [tex]\( x = -3 \)[/tex]; [tex]\( x = -7 \)[/tex]
