To determine the solutions to the quadratic equation [tex]\( x^2 + 4x - 21 = 0 \)[/tex], we need to solve the equation and see which of the given options are correct.
### Step-by-Step Solution:
1. Solve the Quadratic Equation:
The quadratic equation given is [tex]\( x^2 + 4x - 21 = 0 \)[/tex].
We'll start by factoring the quadratic equation.
2. Factoring:
To factor [tex]\( x^2 + 4x - 21 \)[/tex], we need to find two numbers that multiply to [tex]\(-21\)[/tex] (the constant term) and add up to [tex]\(4\)[/tex] (the coefficient of the linear term).
Those two numbers are:
- [tex]\( -3 \)[/tex] and [tex]\( 7 \)[/tex]
Therefore, we can write:
[tex]\[
x^2 + 4x - 21 = (x + 7)(x - 3) = 0
\][/tex]
3. Finding the Roots:
Now that we have factored the quadratic equation, we set each factor to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
(x + 7) = 0 \quad \text{or} \quad (x - 3) = 0
\][/tex]
Solving these gives us:
[tex]\[
x + 7 = 0 \quad \Rightarrow \quad x = -7
\][/tex]
[tex]\[
x - 3 = 0 \quad \Rightarrow \quad x = 3
\][/tex]
So the solutions to the quadratic equation are [tex]\( x = -7 \)[/tex] and [tex]\( x = 3 \)[/tex].
### Checking the Given Options:
Let's see which of the given options match the solutions we found:
- Option A: [tex]\( x = -7 \)[/tex]
This matches our solution.
Correct
- Option B: [tex]\( x = 3 \)[/tex]
This matches our solution.
Correct
- Option C: [tex]\( x = 4 \)[/tex]
This is not one of our solutions.
Incorrect
- Option D: [tex]\( x = 21 \)[/tex]
This is not one of our solutions.
Incorrect
- Option E: [tex]\( x = -5 \)[/tex]
This is not one of our solutions.
Incorrect
### Conclusion:
The correct solutions to the quadratic equation [tex]\( x^2 + 4x - 21 = 0 \)[/tex] that match the given options are:
- Option A: [tex]\( x = -7 \)[/tex]
- Option B: [tex]\( x = 3 \)[/tex]
