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Sagot :
Sure, let's solve the quadratic equation [tex]\(x^2 - (1 + \sqrt{2})x + \sqrt{2} = 0\)[/tex] step by step.
### Step 1: Identify the coefficients
The equation is in the form [tex]\(ax^2 + bx + c = 0\)[/tex]. By comparing, we can identify the coefficients:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = -(1 + \sqrt{2})\)[/tex]
- [tex]\(c = \sqrt{2}\)[/tex]
### Step 2: Calculate the discriminant
The discriminant of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substitute the coefficients into the discriminant formula:
[tex]\[ \Delta = [-(1 + \sqrt{2})]^2 - 4(1)(\sqrt{2}) \][/tex]
[tex]\[ \Delta = (1 + \sqrt{2})^2 - 4\sqrt{2} \][/tex]
[tex]\[ \Delta = 1 + 2\sqrt{2} + 2 - 4\sqrt{2} \][/tex]
[tex]\[ \Delta = 3 - 2\sqrt{2} \][/tex]
The calculated value of the discriminant is approximately:
[tex]\[ \Delta \approx 0.17157287525380926 \][/tex]
### Step 3: Use the quadratic formula to find the solutions
The quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(\Delta\)[/tex] into the formula:
[tex]\[ x = \frac{-( -(1 + \sqrt{2}) ) \pm \sqrt{0.17157287525380926}}{2(1)} \][/tex]
[tex]\[ x = \frac{1 + \sqrt{2} \pm \sqrt{0.17157287525380926}}{2} \][/tex]
### Step 4: Calculate each root
1. For the first root ([tex]\(x_1\)[/tex]):
[tex]\[ x_1 = \frac{1 + \sqrt{2} + \sqrt{0.17157287525380926}}{2} \][/tex]
Approximating the values:
[tex]\[ x_1 \approx \frac{1 + 1.414213562373095 + 0.41421356237309495}{2} \][/tex]
[tex]\[ x_1 \approx \frac{2.82842712474619}{2} \][/tex]
[tex]\[ x_1 \approx 1.4142135623730945 \][/tex]
2. For the second root ([tex]\(x_2\)[/tex]):
[tex]\[ x_2 = \frac{1 + \sqrt{2} - \sqrt{0.17157287525380926}}{2} \][/tex]
Approximating the values:
[tex]\[ x_2 \approx \frac{1 + 1.414213562373095 - 0.41421356237309495}{2} \][/tex]
[tex]\[ x_2 \approx \frac{2.0}{2} \][/tex]
[tex]\[ x_2 \approx 1.0000000000000004 \][/tex]
### Conclusion
Thus, the roots of the quadratic equation [tex]\(x^2 - (1 + \sqrt{2})x + \sqrt{2} = 0\)[/tex] are approximately:
[tex]\[ x_1 \approx 1.4142135623730945 \][/tex]
[tex]\[ x_2 \approx 1.0000000000000004 \][/tex]
And the discriminant is approximately:
[tex]\[ \Delta \approx 0.17157287525380926 \][/tex]
### Step 1: Identify the coefficients
The equation is in the form [tex]\(ax^2 + bx + c = 0\)[/tex]. By comparing, we can identify the coefficients:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = -(1 + \sqrt{2})\)[/tex]
- [tex]\(c = \sqrt{2}\)[/tex]
### Step 2: Calculate the discriminant
The discriminant of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substitute the coefficients into the discriminant formula:
[tex]\[ \Delta = [-(1 + \sqrt{2})]^2 - 4(1)(\sqrt{2}) \][/tex]
[tex]\[ \Delta = (1 + \sqrt{2})^2 - 4\sqrt{2} \][/tex]
[tex]\[ \Delta = 1 + 2\sqrt{2} + 2 - 4\sqrt{2} \][/tex]
[tex]\[ \Delta = 3 - 2\sqrt{2} \][/tex]
The calculated value of the discriminant is approximately:
[tex]\[ \Delta \approx 0.17157287525380926 \][/tex]
### Step 3: Use the quadratic formula to find the solutions
The quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(\Delta\)[/tex] into the formula:
[tex]\[ x = \frac{-( -(1 + \sqrt{2}) ) \pm \sqrt{0.17157287525380926}}{2(1)} \][/tex]
[tex]\[ x = \frac{1 + \sqrt{2} \pm \sqrt{0.17157287525380926}}{2} \][/tex]
### Step 4: Calculate each root
1. For the first root ([tex]\(x_1\)[/tex]):
[tex]\[ x_1 = \frac{1 + \sqrt{2} + \sqrt{0.17157287525380926}}{2} \][/tex]
Approximating the values:
[tex]\[ x_1 \approx \frac{1 + 1.414213562373095 + 0.41421356237309495}{2} \][/tex]
[tex]\[ x_1 \approx \frac{2.82842712474619}{2} \][/tex]
[tex]\[ x_1 \approx 1.4142135623730945 \][/tex]
2. For the second root ([tex]\(x_2\)[/tex]):
[tex]\[ x_2 = \frac{1 + \sqrt{2} - \sqrt{0.17157287525380926}}{2} \][/tex]
Approximating the values:
[tex]\[ x_2 \approx \frac{1 + 1.414213562373095 - 0.41421356237309495}{2} \][/tex]
[tex]\[ x_2 \approx \frac{2.0}{2} \][/tex]
[tex]\[ x_2 \approx 1.0000000000000004 \][/tex]
### Conclusion
Thus, the roots of the quadratic equation [tex]\(x^2 - (1 + \sqrt{2})x + \sqrt{2} = 0\)[/tex] are approximately:
[tex]\[ x_1 \approx 1.4142135623730945 \][/tex]
[tex]\[ x_2 \approx 1.0000000000000004 \][/tex]
And the discriminant is approximately:
[tex]\[ \Delta \approx 0.17157287525380926 \][/tex]
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