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vichka [17]
3 years ago
11

Please help with this ...Its very urgent..

Mathematics
2 answers:
aleksklad [387]3 years ago
8 0
Sorry if it's a little bit ugly

Ad libitum [116K]3 years ago
6 0

Answer: q²x² - p³x + 3pqx + q = 0

<u>Step-by-step explanation:</u>

Since a and B are the roots of x² - px + q = 0, then x = a and x = B

⇒ (x - a)(x - B) = 0

⇒ x² - ax - Bx + aB = 0

⇒ x² - (a + B)x + aB = 0

Comparing this to the given equation of x² - px + q = 0, we discover that

  • p = a + B
  • q = aB

Next, given the roots for the new equation as x = \dfrac{a}{B^2} \quad\text{and}\quad\ x=\dfrac{B}{a^2}

⇒ \bigg(x - \dfrac{a}{B^2}\bigg)\bigg(x-\dfrac{B}{a^2}\bigg)=0

⇒ (B²x - a)(a²x - B) = 0

⇒ a²B²x² - a³x - B³x + aB = 0

⇒ a²B²x² - (a³ + B³)x + aB = 0

Let's look at each term individually:

1st term: a²B²x² = (aB)²x²   = q²x²  <em>(since q = aB)</em>

2nd term: (a³ + B³)x  <em>cubic formula can be used</em>

<em>       </em>        = [(a + B)(a² - aB + B²)]x

              = [(a + B)(a² + (2aB - 3aB) + B²)]x

              = [(a + B)(<u>a² + 2aB + B²</u> - 3aB)]x

              = [(a + B)(a + B)² - 3aB)]x

              = [(   p   )(  p²     -  3 q      )]x      <em>since p = a + B and q = aB</em>

<em>               </em>= (p³ - 3pq)x       <em>distributed "p" into p² - 3q</em>

              = p³x - 3pqx       <em>distributed "x" into p³ - 3pq</em>

3rd term: aB = q   <em>     since q = aB   </em>

Put it all together:

   q²x² - (p³x - 3pqx) + q = 0

⇒ q²x² - p³x + 3pqx + q = 0

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