Answer:
[tex]c^2 = 9dp[/tex]
Step-by-step explanation:
Given
[tex]dx^2 + cx + p = 0[/tex]
Let the roots be [tex]\alpha[/tex] and [tex]\beta[/tex]
So:
[tex]\alpha = 2\beta[/tex]
Required
Determine the relationship between d, c and p
[tex]dx^2 + cx + p = 0[/tex]
Divide through by d
[tex]\frac{dx^2}{d} + \frac{cx}{d} + \frac{p}{d} = 0[/tex]
[tex]x^2 + \frac{c}{d}x + \frac{p}{d} = 0[/tex]
A quadratic equation has the form:
[tex]x^2 - (\alpha + \beta)x + \alpha \beta = 0[/tex]
So:
[tex]x^2 - (2\beta+ \beta)x + \beta*\beta = 0[/tex]
[tex]x^2 - (3\beta)x + \beta^2 = 0[/tex]
So, we have:
[tex]\frac{c}{d} = -3\beta[/tex] -- (1)
and
[tex]\frac{p}{d} = \beta^2[/tex] -- (2)
Make [tex]\beta[/tex] the subject in (1)
[tex]\frac{c}{d} = -3\beta[/tex]
[tex]\beta = -\frac{c}{3d}[/tex]
Substitute [tex]\beta = -\frac{c}{3d}[/tex] in (2)
[tex]\frac{p}{d} = (-\frac{c}{3d})^2[/tex]
[tex]\frac{p}{d} = \frac{c^2}{9d^2}[/tex]
Multiply both sides by d
[tex]d * \frac{p}{d} = \frac{c^2}{9d^2}*d[/tex]
[tex]p = \frac{c^2}{9d}[/tex]
Cross Multiply
[tex]9dp = c^2[/tex]
or
[tex]c^2 = 9dp[/tex]
Hence, the relationship between d, c and p is: [tex]c^2 = 9dp[/tex]
