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QOTD # 20 If a and b are the two solutions of the equation x
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12 Sep 2016, 02:25
12 Sep 2016, 02:25
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If a and b are the two solutions of the equation x² – 5x + 4 = 0, what is the value of \((\frac{1 + a}{a})(\frac{1 + b}{b})\) ?
Give your answer as a fraction.
Give your answer as a fraction.
Show: :: OA
\(\frac{5}{2}\)
Practice Questions
Question: 20
Page: 182
Question: 20
Page: 182
Re: QOTD # 20 If a and b are the two solutions of the equation x
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12 Sep 2016, 02:30
12 Sep 2016, 02:30
2
Expert Reply
Explanation
Factoring the quadratic equation x² – 5x + 4 = 0, you get (x – 1)(x – 4) = 0, so the two solutions are either a = 1 and b = 4 or a = 4 and b = 1. Note that a and b are interchangeable in the expression \(\frac{1 + a}{a} * \frac{1 + b}{b}\) so the value of the expression will be the same regardless of the choices of a and b. Thus, substituting we do have
\(\frac{1 + 1}{1} * \frac{1 + 4}{4} = 2* \frac{5}{4} = \frac{5}{2}\)
Factoring the quadratic equation x² – 5x + 4 = 0, you get (x – 1)(x – 4) = 0, so the two solutions are either a = 1 and b = 4 or a = 4 and b = 1. Note that a and b are interchangeable in the expression \(\frac{1 + a}{a} * \frac{1 + b}{b}\) so the value of the expression will be the same regardless of the choices of a and b. Thus, substituting we do have
\(\frac{1 + 1}{1} * \frac{1 + 4}{4} = 2* \frac{5}{4} = \frac{5}{2}\)
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Re: QOTD # 20 If a and b are the two solutions of the equation x
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02 Jun 2025, 18:42
02 Jun 2025, 18:42
1
\((\frac{1 + a}{a})(\frac{1 + b}{b})\)
=> \(\frac{(1 + a) * (1 + b)}{ab} \)
=> \(\frac{1 + a + b + ab}{ab}\)
a and b are the two solutions of the equation \(x^2 - 5x + 4 = 0\)
We know that for a generic equation \(Ax^2 + Bx + C = 0\)
Sum of roots = \(\frac{-B}{A}\)
Product of roots = \(\frac{C}{A}\)
Comparing \(x^2 - 5x + 4 = 0\) with \(Ax^2 + Bx + C = 0\) we have
A = 1, B = -5, C = 4
=> Sum of Roots a and b = a + b = \(\frac{-B}{A}\) = \(\frac{-(-5)}{1}\) = 5
=> Product of roots a and b = a * b = \(\frac{C}{A}\) = \(\frac{4}{1}\) = 4
=> \(\frac{1 + a + b + ab}{ab}\) = \(\frac{1 + 5 + 4}{4}\) = \(\frac{10}{4}\) = \(\frac{5}{2}\)
So, Answer will be \(\frac{5}{2}\)
Hope it helps!
=> \(\frac{(1 + a) * (1 + b)}{ab} \)
=> \(\frac{1 + a + b + ab}{ab}\)
a and b are the two solutions of the equation \(x^2 - 5x + 4 = 0\)
We know that for a generic equation \(Ax^2 + Bx + C = 0\)
Sum of roots = \(\frac{-B}{A}\)
Product of roots = \(\frac{C}{A}\)
Comparing \(x^2 - 5x + 4 = 0\) with \(Ax^2 + Bx + C = 0\) we have
A = 1, B = -5, C = 4
=> Sum of Roots a and b = a + b = \(\frac{-B}{A}\) = \(\frac{-(-5)}{1}\) = 5
=> Product of roots a and b = a * b = \(\frac{C}{A}\) = \(\frac{4}{1}\) = 4
=> \(\frac{1 + a + b + ab}{ab}\) = \(\frac{1 + 5 + 4}{4}\) = \(\frac{10}{4}\) = \(\frac{5}{2}\)
So, Answer will be \(\frac{5}{2}\)
Hope it helps!
