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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].
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