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Sagot :
Let's solve the quadratic equation step by step to find the solutions for [tex]\(2x^2 + 46x - 8 = 0\)[/tex].
### Step 1: Identify the coefficients
Given the equation [tex]\(2x^2 + 46x - 8 = 0\)[/tex], the coefficients are:
- [tex]\(a = 2\)[/tex]
- [tex]\(b = 46\)[/tex]
- [tex]\(c = -8\)[/tex]
### Step 2: Calculate the discriminant
The discriminant [tex]\(\Delta\)[/tex] of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the values, we get:
[tex]\[ \Delta = 46^2 - 4 \cdot 2 \cdot (-8) = 2116 + 64 = 2180 \][/tex]
### Step 3: Calculate the square root of the discriminant
The square root of the discriminant is:
[tex]\[ \sqrt{\Delta} = \sqrt{2180} \approx 46.69047011971501 \][/tex]
### Step 4: Use the quadratic formula
The solutions [tex]\(x\)[/tex] of the quadratic equation are given by the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substituting the values of [tex]\(b\)[/tex], [tex]\(\sqrt{\Delta}\)[/tex], and [tex]\(a\)[/tex], we get:
[tex]\[ x = \frac{-46 \pm 46.69047011971501}{2 \cdot 2} = \frac{-46 \pm 46.69047011971501}{4} \][/tex]
### Step 5: Solve for the two roots
1. The first solution:
[tex]\[ x_1 = \frac{-46 + 46.69047011971501}{4} = \frac{0.69047011971501}{4} \approx 0.17261752992875223 \][/tex]
2. The second solution:
[tex]\[ x_2 = \frac{-46 - 46.69047011971501}{4} = \frac{-92.69047011971501}{4} \approx -23.17261752992875 \][/tex]
Therefore, the solutions to the quadratic equation [tex]\(2x^2 + 46x - 8 = 0\)[/tex] are approximately [tex]\(0.1726\)[/tex] and [tex]\(-23.1726\)[/tex].
Let's compare these with the options provided. None of the options directly match the numerical solutions, so let's simplify them a bit:
### Simplifying the options:
A. [tex]\( z = 4 \pm 2\sqrt{5} \)[/tex]
B. [tex]\( z = -4 \pm 2\sqrt{5} \)[/tex]
C. [tex]\( z = 2 \pm 4\sqrt{5} \)[/tex]
D. [tex]\( z = -2 \pm 4\sqrt{5} \)[/tex]
Calculating the numerical approximations for each:
- [tex]\(\sqrt{5} \approx 2.236\)[/tex]
For option A:
[tex]\[ z = 4 \pm 2\sqrt{5} \approx 4 \pm 4.472 \implies 8.472 \text{ or } -0.472 \][/tex]
For option B:
[tex]\[ z = -4 \pm 2\sqrt{5} \approx -4 \pm 4.472 \implies 0.472 \text{ or } -8.472 \][/tex]
For option C:
[tex]\[ z = 2 \pm 4\sqrt{5} \approx 2 \pm 8.944 \implies 10.944 \text{ or } -6.944 \][/tex]
For option D:
[tex]\[ z = -2 \pm 4\sqrt{5} \approx -2 \pm 8.944 \implies 6.944 \text{ or } -10.944 \][/tex]
Clearly seeing none of the options A, B, C, or D matches our solutions exactly. The distinct nature of these solutions in the options might suggest different symbolic layers behind the calculations, yet from the numerical accuracy here:
None of the provided choices A, B, C, or D fit the exact numerical solutions [tex]\(0.1726 \text{ and } -23.1726\)[/tex].
### Step 1: Identify the coefficients
Given the equation [tex]\(2x^2 + 46x - 8 = 0\)[/tex], the coefficients are:
- [tex]\(a = 2\)[/tex]
- [tex]\(b = 46\)[/tex]
- [tex]\(c = -8\)[/tex]
### Step 2: Calculate the discriminant
The discriminant [tex]\(\Delta\)[/tex] of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the values, we get:
[tex]\[ \Delta = 46^2 - 4 \cdot 2 \cdot (-8) = 2116 + 64 = 2180 \][/tex]
### Step 3: Calculate the square root of the discriminant
The square root of the discriminant is:
[tex]\[ \sqrt{\Delta} = \sqrt{2180} \approx 46.69047011971501 \][/tex]
### Step 4: Use the quadratic formula
The solutions [tex]\(x\)[/tex] of the quadratic equation are given by the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substituting the values of [tex]\(b\)[/tex], [tex]\(\sqrt{\Delta}\)[/tex], and [tex]\(a\)[/tex], we get:
[tex]\[ x = \frac{-46 \pm 46.69047011971501}{2 \cdot 2} = \frac{-46 \pm 46.69047011971501}{4} \][/tex]
### Step 5: Solve for the two roots
1. The first solution:
[tex]\[ x_1 = \frac{-46 + 46.69047011971501}{4} = \frac{0.69047011971501}{4} \approx 0.17261752992875223 \][/tex]
2. The second solution:
[tex]\[ x_2 = \frac{-46 - 46.69047011971501}{4} = \frac{-92.69047011971501}{4} \approx -23.17261752992875 \][/tex]
Therefore, the solutions to the quadratic equation [tex]\(2x^2 + 46x - 8 = 0\)[/tex] are approximately [tex]\(0.1726\)[/tex] and [tex]\(-23.1726\)[/tex].
Let's compare these with the options provided. None of the options directly match the numerical solutions, so let's simplify them a bit:
### Simplifying the options:
A. [tex]\( z = 4 \pm 2\sqrt{5} \)[/tex]
B. [tex]\( z = -4 \pm 2\sqrt{5} \)[/tex]
C. [tex]\( z = 2 \pm 4\sqrt{5} \)[/tex]
D. [tex]\( z = -2 \pm 4\sqrt{5} \)[/tex]
Calculating the numerical approximations for each:
- [tex]\(\sqrt{5} \approx 2.236\)[/tex]
For option A:
[tex]\[ z = 4 \pm 2\sqrt{5} \approx 4 \pm 4.472 \implies 8.472 \text{ or } -0.472 \][/tex]
For option B:
[tex]\[ z = -4 \pm 2\sqrt{5} \approx -4 \pm 4.472 \implies 0.472 \text{ or } -8.472 \][/tex]
For option C:
[tex]\[ z = 2 \pm 4\sqrt{5} \approx 2 \pm 8.944 \implies 10.944 \text{ or } -6.944 \][/tex]
For option D:
[tex]\[ z = -2 \pm 4\sqrt{5} \approx -2 \pm 8.944 \implies 6.944 \text{ or } -10.944 \][/tex]
Clearly seeing none of the options A, B, C, or D matches our solutions exactly. The distinct nature of these solutions in the options might suggest different symbolic layers behind the calculations, yet from the numerical accuracy here:
None of the provided choices A, B, C, or D fit the exact numerical solutions [tex]\(0.1726 \text{ and } -23.1726\)[/tex].
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