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The roots of \[x^2 + 5x + 3 = 0\]are $p$ and $q,$ and the roots of \[x^2 + bx + c = 0\]are $p^2$ and $q^2.$ Find $b + c.$[tex]The roots of\[x^2 + 5x + 3 = 0\]are $p$ and $q,$ and the roots of\[x^2 + bx + c = 0\]are $p^2$ and $q^2.$ Find $b + c.$[/tex]

Respuesta :

LammettHash

By the factor theorem, we have

[tex]x^2+5x+3=(x-p)(x-q)\implies\begin{cases}3=pq\\-5=p+q\end{cases}[/tex]

[tex]x^2+bx+c=(x-p^2)(x-q^2)\implies\begin{cases}c=p^2q^2\\-b=p^2+q^2\end{cases}[/tex]

Then

[tex]c=p^2q^2=(pq)^2=3^2=9[/tex]

and

[tex]p+q=-5\implies(p+q)^2=p^2+2pq+q^2=25\implies b=-(p^2+q^2)=-19[/tex]

So we have b + c = -19 + 9 = -10.

anuksha0456

The given roots of the equation x² + 5x + 3 = 0 as p and q, and x² + bx + c = 0 as p² and q², gives us the value of b + c = -10.

What are the sum and product of the roots of a quadratic equation?

If the quadratic equation ax² + bx + c = 0, has roots as α and β, then:

  1. Sum of the roots, α + β = -b/a
  2. Product of the roots, αβ = c/a

How to solve the given question?

In the question, we are informed that the roots of the expression x² + 5x + 3 = 0 are p and q, and the roots of the expression x² + bx + c = 0 are p² and q².

We are asked to find the value of b + c.

Since the roots of the equation x² + 5x + 3 = 0 are given as p and q.

So, we can say that:

  1. Sum of the roots, p + q = -5
  2. Product of the roots, pq = 3

Now, we are informed that the roots of the equation x² + bx + c = 0 are p² and q². So, we can say that:

  1. Sum of the roots, p² + q² = -b, or, (p + q)² - 2pq = -b, or, (-5)² - 2(3) = -b, or, 25 - 6 = -b, or, 19 = -b, or, b = -19.
  2. Product of the roots, p²q² = c, or, (pq)² = c, or, 3² = c, or, c = 9.

∴ b + c = -19 + 9 = -10.

∴ The given roots of the equation x² + 5x + 3 = 0 as p and q, and x² + bx + c = 0 as p² and q², gives us the value of b + c = -10.

Learn more about the sum and product of the roots of a quadratic equation at

https://brainly.com/question/10666510

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