It is given the quadratic equation (q + 1 )x² - 8x + p = 0,where p and q are constants,has two equal real roots.Express p in terms of q.
Do help meeee!!!
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Step-by-step explanation:

answer is in the photo above

Answer:

[tex]\displaystyle p = \frac{16}{q + 1}[/tex].

Step-by-step explanation:

Factor out the coefficient of [tex]x^{2}[/tex]:

[tex]\displaystyle \frac{1}{q + 1}\, [(q + 1)\, x^2 - 8\, x + p] = 0[/tex].

[tex]\displaystyle x^2 - \frac{8}{q + 1} \, x + \frac{p}{q + 1} = 0[/tex].

Let [tex]m[/tex] denote the real root of this equation. By the Factor Theorem, this equation would have the factor [tex](x - m)[/tex] repeated for two times in total:

[tex](x - m)^{2} = 0[/tex].

Expand to obtain: [tex]x^2 - 2\, m\cdot x + m^{2} =0[/tex].

Compare this expression with [tex]\displaystyle x^2 - \frac{8}{q + 1} \, x + \frac{p}{q + 1} = 0[/tex]:

[tex]\displaystyle -\frac{8}{q + 1} = -2\, m[/tex] (for the coefficient of [tex]x[/tex].)

[tex]\displaystyle \frac{p}{q + 1} = m^2[/tex] (for the constant.)

Rewrite the first equation to find an expression for [tex]m[/tex] in terms of [tex]q[/tex]:

[tex]\displaystyle m = \frac{4}{q + 1}[/tex].

Substitute this expression for [tex]m[/tex] into the second equation to find an expression for [tex]p[/tex] in terms of [tex]q[/tex]:

[tex]\displaystyle \frac{p}{q + 1} = m^2[/tex].

[tex]\displaystyle \frac{p}{q + 1} = \frac{4^2}{(q+1)^{2}}[/tex].

[tex]\displaystyle p = \frac{16}{q + 1}[/tex].

Verify that this expression for [tex]p[/tex] satisfies the requirements:

[tex]\displaystyle (q + 1) \, x^{2} - 8\, x + \frac{16}{q + 1} = 0[/tex].

[tex]\displaystyle x^2 - \frac{8}{q + 1} + \frac{16}{(q+1)^2} = 0[/tex].

[tex]\displaystyle x = \frac{4}{q + 1}[/tex] is the (repeated) real root of this quadratic equation.

Which of the following is least likely to be an advantage of scientific models?


They can precisely replicate any scientific theory or law.


They can be used to help scientists formulate and improve hypotheses.


They can be used to visually represent objects that are very small or very large.


They can be used to simplify complex concepts so that they are better understood.

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Solve for X

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