IDNLearn.com: Where your questions meet expert advice and community support. Our platform is designed to provide reliable and thorough answers to all your questions, no matter the topic.
Sagot :
To solve the quadratic equation [tex]\(x^2 + 4x - 21 = 0\)[/tex], we follow these steps:
1. Identify the coefficients: The equation is in the standard form [tex]\(ax^2 + bx + c = 0\)[/tex], where [tex]\(a = 1\)[/tex], [tex]\(b = 4\)[/tex], and [tex]\(c = -21\)[/tex].
2. Calculate the discriminant: The discriminant [tex]\(\Delta\)[/tex] is given by the formula [tex]\(b^2 - 4ac\)[/tex].
[tex]\[ \Delta = b^2 - 4ac = 4^2 - 4 \cdot 1 \cdot (-21) = 16 + 84 = 100 \][/tex]
3. Determine the roots using the quadratic formula: The solutions to the equation are found using:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substituting the values [tex]\(a = 1\)[/tex], [tex]\(b = 4\)[/tex], and [tex]\(\Delta = 100\)[/tex], we get:
[tex]\[ x = \frac{-4 \pm \sqrt{100}}{2 \cdot 1} = \frac{-4 \pm 10}{2} \][/tex]
This results in two solutions:
[tex]\[ x_1 = \frac{-4 + 10}{2} = \frac{6}{2} = 3 \][/tex]
[tex]\[ x_2 = \frac{-4 - 10}{2} = \frac{-14}{2} = -7 \][/tex]
4. Verify the solutions: Compare the calculated solutions with the given options:
- Option A: [tex]\(x = -7\)[/tex] matches solution [tex]\(x_2 = -7\)[/tex].
- Option B: [tex]\(x = 3\)[/tex] matches solution [tex]\(x_1 = 3\)[/tex].
- Option C: [tex]\(x = 4\)[/tex] does not match either solution.
- Option D: [tex]\(x = 21\)[/tex] does not match either solution.
- Option E: [tex]\(x = -5\)[/tex] does not match either solution.
Thus, the correct solutions are:
A. [tex]\(x = -7\)[/tex]
B. [tex]\(x = 3\)[/tex]
Therefore, options A and B are correct.
1. Identify the coefficients: The equation is in the standard form [tex]\(ax^2 + bx + c = 0\)[/tex], where [tex]\(a = 1\)[/tex], [tex]\(b = 4\)[/tex], and [tex]\(c = -21\)[/tex].
2. Calculate the discriminant: The discriminant [tex]\(\Delta\)[/tex] is given by the formula [tex]\(b^2 - 4ac\)[/tex].
[tex]\[ \Delta = b^2 - 4ac = 4^2 - 4 \cdot 1 \cdot (-21) = 16 + 84 = 100 \][/tex]
3. Determine the roots using the quadratic formula: The solutions to the equation are found using:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substituting the values [tex]\(a = 1\)[/tex], [tex]\(b = 4\)[/tex], and [tex]\(\Delta = 100\)[/tex], we get:
[tex]\[ x = \frac{-4 \pm \sqrt{100}}{2 \cdot 1} = \frac{-4 \pm 10}{2} \][/tex]
This results in two solutions:
[tex]\[ x_1 = \frac{-4 + 10}{2} = \frac{6}{2} = 3 \][/tex]
[tex]\[ x_2 = \frac{-4 - 10}{2} = \frac{-14}{2} = -7 \][/tex]
4. Verify the solutions: Compare the calculated solutions with the given options:
- Option A: [tex]\(x = -7\)[/tex] matches solution [tex]\(x_2 = -7\)[/tex].
- Option B: [tex]\(x = 3\)[/tex] matches solution [tex]\(x_1 = 3\)[/tex].
- Option C: [tex]\(x = 4\)[/tex] does not match either solution.
- Option D: [tex]\(x = 21\)[/tex] does not match either solution.
- Option E: [tex]\(x = -5\)[/tex] does not match either solution.
Thus, the correct solutions are:
A. [tex]\(x = -7\)[/tex]
B. [tex]\(x = 3\)[/tex]
Therefore, options A and B are correct.
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Your search for answers ends at IDNLearn.com. Thanks for visiting, and we look forward to helping you again soon.