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Sagot :
Let's carefully follow the steps to solve the quadratic equation [tex]\(2x^2 - 7x - 5 = 0\)[/tex] using the quadratic formula. The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
For the given equation, [tex]\(a = 2\)[/tex], [tex]\(b = -7\)[/tex], and [tex]\(c = -5\)[/tex]. Let’s solve it step-by-step while checking each step for correctness:
### Step 1: Calculate the discriminant [tex]\((b^2 - 4ac)\)[/tex]
First, identify the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
- [tex]\(a = 2\)[/tex]
- [tex]\(b = -7\)[/tex]
- [tex]\(c = -5\)[/tex]
Now calculate the discriminant ([tex]\(b^2 - 4ac\)[/tex]):
[tex]\[ b^2 = (-7)^2 = 49 \][/tex]
[tex]\[ 4ac = 4 \cdot 2 \cdot (-5) = -40 \][/tex]
[tex]\[ \text{Discriminant} = 49 - (-40) = 49 + 40 = 89 \][/tex]
So, the discriminant is [tex]\(89\)[/tex].
### Step 2: Calculate the square root of the discriminant and solve for [tex]\(x\)[/tex]
The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Substitute the values of [tex]\(b\)[/tex], [tex]\(a\)[/tex], and the discriminant (computed as [tex]\(89\)[/tex]) into the formula:
[tex]\[ x = \frac{-(-7) \pm \sqrt{89}}{2 \cdot 2} \][/tex]
[tex]\[ x = \frac{7 \pm \sqrt{89}}{4} \][/tex]
### Step 3: Find the roots
Now we find the two possible values for [tex]\(x\)[/tex]:
[tex]\[ x_1 = \frac{7 + \sqrt{89}}{4} \][/tex]
[tex]\[ x_2 = \frac{7 - \sqrt{89}}{4} \][/tex]
Simplify these expressions further:
[tex]\[ x_1 = 4.10849528301415 \][/tex]
[tex]\[ x_2 = -0.6084952830141508 \][/tex]
### Reviewing Nisha's Steps:
- Nisha's Step 1:
[tex]\[ x = \frac{7 \pm \sqrt{49 - 40}}{4} \][/tex]
[tex]\[ x = \frac{7 \pm \sqrt{9}}{4} \][/tex]
Nisha subtracts and simplifies the discriminant incorrectly. The correct discriminant should be [tex]\(89\)[/tex] and not [tex]\(9\)[/tex]. So, she made an error in this step by subtracting [tex]\(40\)[/tex] instead of adding it.
- Nisha's Step 2:
[tex]\[ x = \frac{7 \pm \sqrt{9}}{4} \][/tex]
Given her incorrect discriminant, she calculates the square root of [tex]\(9\)[/tex] correctly but this step is already invalid.
- Nisha's Step 3:
[tex]\[ x = \left\{\frac{5}{2}, 1\right\} \][/tex]
Here, Nisha derives incorrect roots from an invalid discriminant calculation.
Thus, Nisha should have added [tex]\(40\)[/tex] instead of subtracting it in her first step.
### Conclusion:
Nisha made the error in Step 1. She should have added [tex]\(40\)[/tex] underneath the radical instead of subtracting it. The accurate roots are [tex]\(4.10849528301415\)[/tex] and [tex]\(-0.6084952830141508\)[/tex].
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
For the given equation, [tex]\(a = 2\)[/tex], [tex]\(b = -7\)[/tex], and [tex]\(c = -5\)[/tex]. Let’s solve it step-by-step while checking each step for correctness:
### Step 1: Calculate the discriminant [tex]\((b^2 - 4ac)\)[/tex]
First, identify the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
- [tex]\(a = 2\)[/tex]
- [tex]\(b = -7\)[/tex]
- [tex]\(c = -5\)[/tex]
Now calculate the discriminant ([tex]\(b^2 - 4ac\)[/tex]):
[tex]\[ b^2 = (-7)^2 = 49 \][/tex]
[tex]\[ 4ac = 4 \cdot 2 \cdot (-5) = -40 \][/tex]
[tex]\[ \text{Discriminant} = 49 - (-40) = 49 + 40 = 89 \][/tex]
So, the discriminant is [tex]\(89\)[/tex].
### Step 2: Calculate the square root of the discriminant and solve for [tex]\(x\)[/tex]
The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Substitute the values of [tex]\(b\)[/tex], [tex]\(a\)[/tex], and the discriminant (computed as [tex]\(89\)[/tex]) into the formula:
[tex]\[ x = \frac{-(-7) \pm \sqrt{89}}{2 \cdot 2} \][/tex]
[tex]\[ x = \frac{7 \pm \sqrt{89}}{4} \][/tex]
### Step 3: Find the roots
Now we find the two possible values for [tex]\(x\)[/tex]:
[tex]\[ x_1 = \frac{7 + \sqrt{89}}{4} \][/tex]
[tex]\[ x_2 = \frac{7 - \sqrt{89}}{4} \][/tex]
Simplify these expressions further:
[tex]\[ x_1 = 4.10849528301415 \][/tex]
[tex]\[ x_2 = -0.6084952830141508 \][/tex]
### Reviewing Nisha's Steps:
- Nisha's Step 1:
[tex]\[ x = \frac{7 \pm \sqrt{49 - 40}}{4} \][/tex]
[tex]\[ x = \frac{7 \pm \sqrt{9}}{4} \][/tex]
Nisha subtracts and simplifies the discriminant incorrectly. The correct discriminant should be [tex]\(89\)[/tex] and not [tex]\(9\)[/tex]. So, she made an error in this step by subtracting [tex]\(40\)[/tex] instead of adding it.
- Nisha's Step 2:
[tex]\[ x = \frac{7 \pm \sqrt{9}}{4} \][/tex]
Given her incorrect discriminant, she calculates the square root of [tex]\(9\)[/tex] correctly but this step is already invalid.
- Nisha's Step 3:
[tex]\[ x = \left\{\frac{5}{2}, 1\right\} \][/tex]
Here, Nisha derives incorrect roots from an invalid discriminant calculation.
Thus, Nisha should have added [tex]\(40\)[/tex] instead of subtracting it in her first step.
### Conclusion:
Nisha made the error in Step 1. She should have added [tex]\(40\)[/tex] underneath the radical instead of subtracting it. The accurate roots are [tex]\(4.10849528301415\)[/tex] and [tex]\(-0.6084952830141508\)[/tex].
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