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Sagot :
To solve the quadratic equation [tex]\(x^2 - 16x + 64 = 0\)[/tex], we follow these steps:
### Step 1: Identify coefficients
The quadratic equation is in the standard form [tex]\(ax^2 + bx + c = 0\)[/tex], where:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = -16\)[/tex]
- [tex]\(c = 64\)[/tex]
### Step 2: Calculate the discriminant
The discriminant [tex]\(\Delta\)[/tex] is given by the formula:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ \Delta = (-16)^2 - 4 \cdot 1 \cdot 64 \][/tex]
[tex]\[ \Delta = 256 - 256 \][/tex]
[tex]\[ \Delta = 0 \][/tex]
### Step 3: Determine the nature of the roots
Since the discriminant [tex]\(\Delta\)[/tex] is 0, this indicates that the quadratic equation has exactly one real root (a repeated root).
### Step 4: Calculate the root(s)
The roots of the quadratic equation are given by the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Since [tex]\(\Delta = 0\)[/tex], the formula simplifies to:
[tex]\[ x = \frac{-b}{2a} \][/tex]
Substitute the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[ x = \frac{-(-16)}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{16}{2} \][/tex]
[tex]\[ x = 8 \][/tex]
Therefore, the quadratic equation [tex]\(x^2 - 16x + 64 = 0\)[/tex] has a double root at [tex]\(x = 8\)[/tex].
### Step 5: Choose the correct answer
From the given options:
- A. [tex]\(x = 4\)[/tex] and [tex]\(x = 16\)[/tex]
- B. [tex]\(x = 8\)[/tex] only
- C. [tex]\(x = 8\)[/tex] and [tex]\(x = -8\)[/tex]
- D. [tex]\(x = -2\)[/tex] and [tex]\(x = 32\)[/tex]
The correct answer is B. [tex]\(x = 8\)[/tex] only.
### Step 1: Identify coefficients
The quadratic equation is in the standard form [tex]\(ax^2 + bx + c = 0\)[/tex], where:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = -16\)[/tex]
- [tex]\(c = 64\)[/tex]
### Step 2: Calculate the discriminant
The discriminant [tex]\(\Delta\)[/tex] is given by the formula:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ \Delta = (-16)^2 - 4 \cdot 1 \cdot 64 \][/tex]
[tex]\[ \Delta = 256 - 256 \][/tex]
[tex]\[ \Delta = 0 \][/tex]
### Step 3: Determine the nature of the roots
Since the discriminant [tex]\(\Delta\)[/tex] is 0, this indicates that the quadratic equation has exactly one real root (a repeated root).
### Step 4: Calculate the root(s)
The roots of the quadratic equation are given by the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Since [tex]\(\Delta = 0\)[/tex], the formula simplifies to:
[tex]\[ x = \frac{-b}{2a} \][/tex]
Substitute the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[ x = \frac{-(-16)}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{16}{2} \][/tex]
[tex]\[ x = 8 \][/tex]
Therefore, the quadratic equation [tex]\(x^2 - 16x + 64 = 0\)[/tex] has a double root at [tex]\(x = 8\)[/tex].
### Step 5: Choose the correct answer
From the given options:
- A. [tex]\(x = 4\)[/tex] and [tex]\(x = 16\)[/tex]
- B. [tex]\(x = 8\)[/tex] only
- C. [tex]\(x = 8\)[/tex] and [tex]\(x = -8\)[/tex]
- D. [tex]\(x = -2\)[/tex] and [tex]\(x = 32\)[/tex]
The correct answer is B. [tex]\(x = 8\)[/tex] only.
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