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Example 1
Find the values of the coefficients $b$ and $c$ in the quadratic equation $2 x^2+b x+c$ if the sum and product of the roots of this equation are respectively $2\dfrac{1}{2}$ and $3\dfrac{1}{4}$.
- $b= 2\dfrac{1}{2} , c = 3\dfrac{1}{4}$
- $b= 6\dfrac{1}{2} , c = -5 $
- $b= 3\dfrac{1}{4} , c = 2\dfrac{1}{2} $
- $b= -5 , c = 6\dfrac{1}{2}$
- $b=-2 , c = -2$
Solution
- The sum $s$ and the product $p$ of the roots of a quadratic equation of the form $a x^2+b x+c$ are given by
$s=\dfrac{-b}{a}$ and $p=\dfrac{c}{a}$
- The coefficient $a$ in the given equation is known. Hence
$s=2\dfrac{1}{2}=\dfrac{-b}{2}$ and $p=3\dfrac{1}{4}=\dfrac{c}{2}$
- Convert the left side of each equation into a fraction and rewrite the above equations as follows
$\dfrac{5}{2}=\dfrac{-b}{2}$ and $\dfrac{13}{4}=\dfrac{c}{2}$
- Solve for $b$ and $c$: $b=-5$ and $c=6\dfrac{1}{2}$
Answer D
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