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Sagot :
To determine the minimum value of [tex]\( m \)[/tex] given the quadratic equation [tex]\( x^2 - mx + 121 = 0 \)[/tex], where both roots are greater than 10, let's follow these detailed steps:
1. Identify the form of the quadratic equation:
The equation is [tex]\( x^2 - mx + 121 = 0 \)[/tex] where [tex]\( m \)[/tex] is the coefficient of [tex]\( x \)[/tex].
2. Understand the conditions for the roots:
We are given that both roots of the equation are greater than 10.
3. Using the properties of quadratic equations:
For a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex], the sum of the roots ([tex]\(\alpha + \beta\)[/tex]) and the product of the roots ([tex]\(\alpha \beta\)[/tex]) are given by:
- Sum of the roots: [tex]\( \alpha + \beta = -\frac{b}{a} \)[/tex]
- Product of the roots: [tex]\( \alpha \beta = \frac{c}{a} \)[/tex]
For our specific equation [tex]\( x^2 - mx + 121 = 0 \)[/tex]:
- [tex]\( a = 1 \)[/tex], [tex]\( b = -m \)[/tex], [tex]\( c = 121 \)[/tex]
- Thus, the sum of the roots [tex]\( \alpha + \beta = m \)[/tex] and the product of the roots [tex]\( \alpha \beta = 121 \)[/tex].
4. Set up inequalities based on the given conditions:
Given that both roots are greater than 10, we can denote the roots as [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex] where [tex]\(\alpha > 10\)[/tex] and [tex]\(\beta > 10\)[/tex].
5. Find suitable roots:
Let's satisfy the conditions given:
- Assume the roots [tex]\( \alpha = 10 \)[/tex] and [tex]\(\beta = 12.1\)[/tex] (as both are greater than 10 and their product is close to 121).
6. Calculate the sum of the roots to find [tex]\( m \)[/tex]:
- [tex]\( \alpha + \beta = 10 + 12.1 = 22.1 \)[/tex]
Thus, the minimum value of [tex]\( m \)[/tex] such that both roots of the quadratic equation [tex]\( x^2 - mx + 121 = 0 \)[/tex] are greater than 10 is [tex]\( \boxed{22.1} \)[/tex].
Since this value doesn't exactly match one of the multiple-choice options, none of the given options is correct. Therefore, the correct response to the question based on the options is:
(d) Cannot be determined
1. Identify the form of the quadratic equation:
The equation is [tex]\( x^2 - mx + 121 = 0 \)[/tex] where [tex]\( m \)[/tex] is the coefficient of [tex]\( x \)[/tex].
2. Understand the conditions for the roots:
We are given that both roots of the equation are greater than 10.
3. Using the properties of quadratic equations:
For a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex], the sum of the roots ([tex]\(\alpha + \beta\)[/tex]) and the product of the roots ([tex]\(\alpha \beta\)[/tex]) are given by:
- Sum of the roots: [tex]\( \alpha + \beta = -\frac{b}{a} \)[/tex]
- Product of the roots: [tex]\( \alpha \beta = \frac{c}{a} \)[/tex]
For our specific equation [tex]\( x^2 - mx + 121 = 0 \)[/tex]:
- [tex]\( a = 1 \)[/tex], [tex]\( b = -m \)[/tex], [tex]\( c = 121 \)[/tex]
- Thus, the sum of the roots [tex]\( \alpha + \beta = m \)[/tex] and the product of the roots [tex]\( \alpha \beta = 121 \)[/tex].
4. Set up inequalities based on the given conditions:
Given that both roots are greater than 10, we can denote the roots as [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex] where [tex]\(\alpha > 10\)[/tex] and [tex]\(\beta > 10\)[/tex].
5. Find suitable roots:
Let's satisfy the conditions given:
- Assume the roots [tex]\( \alpha = 10 \)[/tex] and [tex]\(\beta = 12.1\)[/tex] (as both are greater than 10 and their product is close to 121).
6. Calculate the sum of the roots to find [tex]\( m \)[/tex]:
- [tex]\( \alpha + \beta = 10 + 12.1 = 22.1 \)[/tex]
Thus, the minimum value of [tex]\( m \)[/tex] such that both roots of the quadratic equation [tex]\( x^2 - mx + 121 = 0 \)[/tex] are greater than 10 is [tex]\( \boxed{22.1} \)[/tex].
Since this value doesn't exactly match one of the multiple-choice options, none of the given options is correct. Therefore, the correct response to the question based on the options is:
(d) Cannot be determined
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