The sum of a number and its square is \(\frac{63}{4}\), Find the numbers.
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Let the number be x.
So, its square will be x2.
From the question, it’s given that sum of the number and its square is \(\frac{63}{4}\)
Which means,
x + x2 = \(\frac{63}{4}\)
⇒ 4x + 4x2 = 63
⇒ 4x2 + 4x – 63 = 0
Solving for x by factorization method, we have
⇒ 4x2 + 18x – 14x – 63 = 0
⇒ 2x(2x + 9) – 7(2x – 9) = 0
⇒ (2x – 7)(2x + 9) = 0
Now, either 2x -7 = 0 ⇒ x = \(\frac{7}{2}\)
Or, 2x + 9 = 0 ⇒ x = \(\frac{-9}{2}\)
Thus, the numbers are \(\frac{7}{2}\) and \(\frac{-9}{2}\).