Connect with knowledgeable experts and enthusiasts on IDNLearn.com. Explore thousands of verified answers from experts and find the solutions you need, no matter the topic.
Sagot :
Let's solve the quadratic equation [tex]\(3x^2 + 24x - 24 = 0\)[/tex] step by step to determine the correct answer among the given choices.
### Step 1: Identify the coefficients
For the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex]:
- [tex]\(a = 3\)[/tex]
- [tex]\(b = 24\)[/tex]
- [tex]\(c = -24\)[/tex]
### Step 2: Calculate the discriminant
The discriminant ([tex]\(\Delta\)[/tex]) of a quadratic equation is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the known values:
[tex]\[ \Delta = 24^2 - 4 \cdot 3 \cdot (-24) \][/tex]
Calculate each term:
[tex]\[ 24^2 = 576 \][/tex]
[tex]\[ 4 \cdot 3 \cdot (-24) = -288 \cdot (-1) = 288 \][/tex]
Summing these values:
[tex]\[ \Delta = 576 + 288 = 864 \][/tex]
### Step 3: Calculate the roots using the quadratic formula
The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(\Delta\)[/tex]:
[tex]\[ x = \frac{-24 \pm \sqrt{864}}{2 \cdot 3} \][/tex]
Simplify:
[tex]\[ x = \frac{-24 \pm \sqrt{864}}{6} \][/tex]
### Step 4: Simplify the square root
Note that:
[tex]\[ 864 = 144 \times 6 = (12)^2 \times 6 \][/tex]
So,
[tex]\[ \sqrt{864} = \sqrt{(12)^2 \times 6} = 12\sqrt{6} \][/tex]
Now substitute back in:
[tex]\[ x = \frac{-24 \pm 12\sqrt{6}}{6} \][/tex]
### Step 5: Simplify the fraction
Divide both terms in the numerator by 6:
[tex]\[ x = -4 \pm 2\sqrt{6} \][/tex]
Therefore, the solutions are:
[tex]\[ x = -4 + 2\sqrt{6} \quad \text{and} \quad x = -4 - 2\sqrt{6} \][/tex]
### Conclusion
Comparing these solutions with the options provided:
- Option A: [tex]\(x = -2 \pm 4\sqrt{6}\)[/tex]
- Option B: [tex]\(x = -4 \pm 2\sqrt{6}\)[/tex]
- Option C: [tex]\(x = 2 \pm 4\sqrt{6}\)[/tex]
- Option D: [tex]\(x = 4 \pm 2\sqrt{6}\)[/tex]
The correct answer is:
[tex]\[ \boxed{\text{B: } x = -4 \pm 2\sqrt{6}} \][/tex]
The correct selection based on the given solutions and calculations is: [tex]\(B: x = -4 \pm 2\sqrt{6}\)[/tex].
### Step 1: Identify the coefficients
For the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex]:
- [tex]\(a = 3\)[/tex]
- [tex]\(b = 24\)[/tex]
- [tex]\(c = -24\)[/tex]
### Step 2: Calculate the discriminant
The discriminant ([tex]\(\Delta\)[/tex]) of a quadratic equation is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the known values:
[tex]\[ \Delta = 24^2 - 4 \cdot 3 \cdot (-24) \][/tex]
Calculate each term:
[tex]\[ 24^2 = 576 \][/tex]
[tex]\[ 4 \cdot 3 \cdot (-24) = -288 \cdot (-1) = 288 \][/tex]
Summing these values:
[tex]\[ \Delta = 576 + 288 = 864 \][/tex]
### Step 3: Calculate the roots using the quadratic formula
The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(\Delta\)[/tex]:
[tex]\[ x = \frac{-24 \pm \sqrt{864}}{2 \cdot 3} \][/tex]
Simplify:
[tex]\[ x = \frac{-24 \pm \sqrt{864}}{6} \][/tex]
### Step 4: Simplify the square root
Note that:
[tex]\[ 864 = 144 \times 6 = (12)^2 \times 6 \][/tex]
So,
[tex]\[ \sqrt{864} = \sqrt{(12)^2 \times 6} = 12\sqrt{6} \][/tex]
Now substitute back in:
[tex]\[ x = \frac{-24 \pm 12\sqrt{6}}{6} \][/tex]
### Step 5: Simplify the fraction
Divide both terms in the numerator by 6:
[tex]\[ x = -4 \pm 2\sqrt{6} \][/tex]
Therefore, the solutions are:
[tex]\[ x = -4 + 2\sqrt{6} \quad \text{and} \quad x = -4 - 2\sqrt{6} \][/tex]
### Conclusion
Comparing these solutions with the options provided:
- Option A: [tex]\(x = -2 \pm 4\sqrt{6}\)[/tex]
- Option B: [tex]\(x = -4 \pm 2\sqrt{6}\)[/tex]
- Option C: [tex]\(x = 2 \pm 4\sqrt{6}\)[/tex]
- Option D: [tex]\(x = 4 \pm 2\sqrt{6}\)[/tex]
The correct answer is:
[tex]\[ \boxed{\text{B: } x = -4 \pm 2\sqrt{6}} \][/tex]
The correct selection based on the given solutions and calculations is: [tex]\(B: x = -4 \pm 2\sqrt{6}\)[/tex].
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. IDNLearn.com has the solutions you’re looking for. Thanks for visiting, and see you next time for more reliable information.