Answer :
To solve the quadratic equation [tex]\( x^2 + 4x - 21 = 0 \)[/tex], we can use the quadratic formula, which is given by:
[tex]\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \][/tex]
Here, the coefficients are:
[tex]\[ a = 1 \][/tex]
[tex]\[ b = 4 \][/tex]
[tex]\[ c = -21 \][/tex]
### Step-by-step solution:
1. Calculate the discriminant:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
[tex]\[ \Delta = 4^2 - 4 \cdot 1 \cdot (-21) \][/tex]
[tex]\[ \Delta = 16 + 84 \][/tex]
[tex]\[ \Delta = 100 \][/tex]
2. Calculate the two potential solutions using the quadratic formula:
[tex]\[ x_1 = \frac{{-b + \sqrt{\Delta}}}{2a} \quad \text{and} \quad x_2 = \frac{{-b - \sqrt{\Delta}}}{2a} \][/tex]
Substituting the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( \Delta \)[/tex]:
[tex]\[ x_1 = \frac{{-4 + \sqrt{100}}}{2 \cdot 1} \][/tex]
[tex]\[ x_1 = \frac{{-4 + 10}}{2} \][/tex]
[tex]\[ x_1 = \frac{6}{2} \][/tex]
[tex]\[ x_1 = 3 \][/tex]
Similarly,
[tex]\[ x_2 = \frac{{-4 - \sqrt{100}}}{2 \cdot 1} \][/tex]
[tex]\[ x_2 = \frac{{-4 - 10}}{2} \][/tex]
[tex]\[ x_2 = \frac{-14}{2} \][/tex]
[tex]\[ x_2 = -7 \][/tex]
### Conclusion:
The solutions to the quadratic equation [tex]\( x^2 + 4x - 21 = 0 \)[/tex] are [tex]\( x = 3 \)[/tex] and [tex]\( x = -7 \)[/tex].
Therefore, the correct answer is:
C. [tex]\( 3, -7 \)[/tex]
[tex]\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \][/tex]
Here, the coefficients are:
[tex]\[ a = 1 \][/tex]
[tex]\[ b = 4 \][/tex]
[tex]\[ c = -21 \][/tex]
### Step-by-step solution:
1. Calculate the discriminant:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
[tex]\[ \Delta = 4^2 - 4 \cdot 1 \cdot (-21) \][/tex]
[tex]\[ \Delta = 16 + 84 \][/tex]
[tex]\[ \Delta = 100 \][/tex]
2. Calculate the two potential solutions using the quadratic formula:
[tex]\[ x_1 = \frac{{-b + \sqrt{\Delta}}}{2a} \quad \text{and} \quad x_2 = \frac{{-b - \sqrt{\Delta}}}{2a} \][/tex]
Substituting the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( \Delta \)[/tex]:
[tex]\[ x_1 = \frac{{-4 + \sqrt{100}}}{2 \cdot 1} \][/tex]
[tex]\[ x_1 = \frac{{-4 + 10}}{2} \][/tex]
[tex]\[ x_1 = \frac{6}{2} \][/tex]
[tex]\[ x_1 = 3 \][/tex]
Similarly,
[tex]\[ x_2 = \frac{{-4 - \sqrt{100}}}{2 \cdot 1} \][/tex]
[tex]\[ x_2 = \frac{{-4 - 10}}{2} \][/tex]
[tex]\[ x_2 = \frac{-14}{2} \][/tex]
[tex]\[ x_2 = -7 \][/tex]
### Conclusion:
The solutions to the quadratic equation [tex]\( x^2 + 4x - 21 = 0 \)[/tex] are [tex]\( x = 3 \)[/tex] and [tex]\( x = -7 \)[/tex].
Therefore, the correct answer is:
C. [tex]\( 3, -7 \)[/tex]
