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Sagot :
Let's break this down step by step.
First, we are given that the original quadratic equation [tex]\( f(x) = 4x^2 - 3x - 5 \)[/tex] has roots [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex].
### Part (a)
We need to form a new equation that has roots [tex]\(\frac{2\alpha}{\beta}\)[/tex] and [tex]\(\frac{2\beta}{\alpha}\)[/tex].
1. Sum of the new roots:
[tex]\[ S = \frac{2\alpha}{\beta} + \frac{2\beta}{\alpha} \][/tex]
2. Product of the new roots:
[tex]\[ P = \left( \frac{2\alpha}{\beta} \right) \left( \frac{2\beta}{\alpha} \right) = 4 \][/tex]
Thus, we can write the quadratic equation with these roots as:
[tex]\[ g(x) = x^2 - Sx + P \][/tex]
Substituting the sum and the product of the new roots:
[tex]\[ S = \frac{2\alpha}{\beta} + \frac{2\beta}{\alpha} \][/tex]
The simplified form of the sum [tex]\(\frac{2\alpha}{\beta} + \frac{2\beta}{\alpha}\)[/tex] is simply [tex]\(S\)[/tex].
Thus
[tex]\[ g(x) = x^2 - \left( \frac{2\alpha}{\beta} + \frac{2\beta}{\alpha} \right)x + 4 \][/tex]
Given that the new equation has integer coefficients and should be of the form [tex]\(g(x)=4x^2+px+q\)[/tex], we need to multiply the entire equation by 4
[tex]\[ 4g(x) = 4x^2 - 4\left(\frac{2\alpha}{\beta} + \frac{2\beta}{\alpha}\right)x + 16 \][/tex]
Thus the simplified form is:
[tex]\[ g(x) = 4x^2 - \left(4\frac{2\alpha}{\beta} + 4\frac{2\beta}{\alpha}\right)x + 16 \][/tex]
Now we have:
[tex]\[ p = 4\left( \frac{2\alpha}{\beta} + \frac{2\beta}{\alpha} \right) x \quad \text{and} \quad q = 16 \][/tex]
### Part (b)
We also know that this new equation [tex]\( g(x) = 4x^2 + px + q \)[/tex] has roots [tex]\( 3\alpha + \beta \)[/tex] and [tex]\( \alpha + 3\beta \)[/tex].
1. Sum of these new roots:
[tex]\[ p = -( (3\alpha + \beta) + (\alpha + 3\beta) ) = -4(\alpha + \beta) \][/tex]
2. Product of these new roots:
[tex]\[ q = (3\alpha + \beta)(\alpha + 3\beta) \][/tex]
Here,
[tex]\[ p = -4(\alpha + \beta) \][/tex]
And solving the product:
[tex]\[ q = (\alpha + 3\beta)(3\alpha + \beta) = 3\alpha^2 + 3\alpha\beta + \beta\alpha + 3\beta^2 \][/tex]
Rewritten:
[tex]\[ q = (3\alpha \beta+\alpha +3(3\alpha \beta) + \beta) \][/tex]
We can conclude resulting the product:
[tex]\[ q = (\alpha + 3\beta) *(\beta + 3\alpha) = (\alpha \beta)^2 + 6 \alpha \beta + 9 \alpha \beta = \][/tex]
So the coefficients of [tex]\( g(x) = 4x^2 + px + q \)[/tex] are:
[tex]\[ p = -4\alpha - 4\beta \quad ,q = (\alpha+3\beta)(3\alpha+\beta) \][/tex]
In short, the values of [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are:
[tex]\[ p = -4(\alpha + \beta), \quad q = (\alpha + 3\beta)(3\alpha + \beta) \][/tex]
First, we are given that the original quadratic equation [tex]\( f(x) = 4x^2 - 3x - 5 \)[/tex] has roots [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex].
### Part (a)
We need to form a new equation that has roots [tex]\(\frac{2\alpha}{\beta}\)[/tex] and [tex]\(\frac{2\beta}{\alpha}\)[/tex].
1. Sum of the new roots:
[tex]\[ S = \frac{2\alpha}{\beta} + \frac{2\beta}{\alpha} \][/tex]
2. Product of the new roots:
[tex]\[ P = \left( \frac{2\alpha}{\beta} \right) \left( \frac{2\beta}{\alpha} \right) = 4 \][/tex]
Thus, we can write the quadratic equation with these roots as:
[tex]\[ g(x) = x^2 - Sx + P \][/tex]
Substituting the sum and the product of the new roots:
[tex]\[ S = \frac{2\alpha}{\beta} + \frac{2\beta}{\alpha} \][/tex]
The simplified form of the sum [tex]\(\frac{2\alpha}{\beta} + \frac{2\beta}{\alpha}\)[/tex] is simply [tex]\(S\)[/tex].
Thus
[tex]\[ g(x) = x^2 - \left( \frac{2\alpha}{\beta} + \frac{2\beta}{\alpha} \right)x + 4 \][/tex]
Given that the new equation has integer coefficients and should be of the form [tex]\(g(x)=4x^2+px+q\)[/tex], we need to multiply the entire equation by 4
[tex]\[ 4g(x) = 4x^2 - 4\left(\frac{2\alpha}{\beta} + \frac{2\beta}{\alpha}\right)x + 16 \][/tex]
Thus the simplified form is:
[tex]\[ g(x) = 4x^2 - \left(4\frac{2\alpha}{\beta} + 4\frac{2\beta}{\alpha}\right)x + 16 \][/tex]
Now we have:
[tex]\[ p = 4\left( \frac{2\alpha}{\beta} + \frac{2\beta}{\alpha} \right) x \quad \text{and} \quad q = 16 \][/tex]
### Part (b)
We also know that this new equation [tex]\( g(x) = 4x^2 + px + q \)[/tex] has roots [tex]\( 3\alpha + \beta \)[/tex] and [tex]\( \alpha + 3\beta \)[/tex].
1. Sum of these new roots:
[tex]\[ p = -( (3\alpha + \beta) + (\alpha + 3\beta) ) = -4(\alpha + \beta) \][/tex]
2. Product of these new roots:
[tex]\[ q = (3\alpha + \beta)(\alpha + 3\beta) \][/tex]
Here,
[tex]\[ p = -4(\alpha + \beta) \][/tex]
And solving the product:
[tex]\[ q = (\alpha + 3\beta)(3\alpha + \beta) = 3\alpha^2 + 3\alpha\beta + \beta\alpha + 3\beta^2 \][/tex]
Rewritten:
[tex]\[ q = (3\alpha \beta+\alpha +3(3\alpha \beta) + \beta) \][/tex]
We can conclude resulting the product:
[tex]\[ q = (\alpha + 3\beta) *(\beta + 3\alpha) = (\alpha \beta)^2 + 6 \alpha \beta + 9 \alpha \beta = \][/tex]
So the coefficients of [tex]\( g(x) = 4x^2 + px + q \)[/tex] are:
[tex]\[ p = -4\alpha - 4\beta \quad ,q = (\alpha+3\beta)(3\alpha+\beta) \][/tex]
In short, the values of [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are:
[tex]\[ p = -4(\alpha + \beta), \quad q = (\alpha + 3\beta)(3\alpha + \beta) \][/tex]
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