Respuesta :
Answer:
6x² + 5x - 2 = 0
Step-by-step explanation:
Given
2x² - 5x - 6 = 0 ← in standard form
with a = 2, b = - 5, c = - 6, then
sum of roots α + β = - [tex]\frac{b}{a}[/tex] = [tex]\frac{5}{2}[/tex]
product of roots = [tex]\frac{c}{a}[/tex] = - 3, then
sum of new roots = [tex]\frac{1}{\alpha }[/tex] + [tex]\frac{1}{\beta }[/tex]
= [tex]\frac{\beta+\alpha }{\alpha \beta }[/tex]
= [tex]\frac{\frac{5}{2} }{-3}[/tex] = - [tex]\frac{5}{6}[/tex]
product of new roots = [tex]\frac{1}{\alpha }[/tex] × [tex]\frac{1}{\beta }[/tex]
= [tex]\frac{1}{\alpha\beta }[/tex] = - [tex]\frac{1}{3}[/tex]
Hence the required equation is
x² + [tex]\frac{5}{6}[/tex] x - [tex]\frac{1}{3}[/tex] = 0 or
6x² + 5x - 2 = 0 ( multiplying through by 6 )
A quadratic equation is represented as: [tex]\mathbf{x^2 - (sum\ of\ roots)x + (product\ of\ roots) = 0}[/tex].
The required quadratic equation is:[tex]\mathbf{x^2 + \frac 56x - \frac{1}{3} = 0}[/tex]
The equation is given as:
[tex]\mathbf{2x^2 - 5x - 6 = 0}[/tex]
Divide through by 2
[tex]\mathbf{x^2 - \frac{5}{2}x - 3 = 0}[/tex]
So, we have:
[tex]\mathbf{\alpha + \beta = -(-\frac{5}{2})}[/tex]
[tex]\mathbf{\alpha + \beta = \frac{5}{2}}[/tex] --- (1)
And:
[tex]\mathbf{\alpha \beta = -3}[/tex] --- (2)
Divide (1) by (2)
[tex]\mathbf{\frac{\alpha + \beta}{\alpha \beta} = \frac{5}{2} \div -3}[/tex]
[tex]\mathbf{\frac{\alpha + \beta}{\alpha \beta} =- \frac{5}{6}}[/tex]
Split
[tex]\mathbf{\frac{\alpha}{\alpha \beta}+ \frac{\beta}{\alpha \beta} =- \frac{5}{6}}[/tex]
[tex]\mathbf{\frac{1}{\beta}+ \frac{1}{\alpha } =- \frac{5}{6}}[/tex]
Take inverse of [tex]\mathbf{\alpha \beta = -3}[/tex]
[tex]\mathbf{\frac{1}{\alpha \beta }= -\frac 13}[/tex]
So, the equation with roots 1/α and 1/β is:
[tex]\mathbf{x^2 - (\frac{1}{\beta}+ \frac{1}{\alpha })x + \frac{1}{\alpha \beta } = 0}[/tex]
This gives
[tex]\mathbf{x^2 - (-\frac 56)x - \frac{1}{3} = 0}[/tex]
[tex]\mathbf{x^2 + \frac 56x - \frac{1}{3} = 0}[/tex]
Hence, the quadratic equation is:[tex]\mathbf{x^2 + \frac 56x - \frac{1}{3} = 0}[/tex]
Read more about quadratic equations at:
https://brainly.com/question/17788108
