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Math Help - Quadratic Eq.

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    Member ~berserk's Avatar
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    Quadratic Eq.

    Write the quadratic equation such that the sum of its roots is -5 and the product of its roots is 6 I know that sum=-b/a and product=c/a
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    Quote Originally Posted by ~berserk View Post
    Write the quadratic equation such that the sum of its roots is -5 and the product of its roots is 6 I know that sum=-b/a and product=c/a
    (x - 2)(x - 3)

    x^2 -5x + 6 = 0
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    Hi, berserk! The easiest way to solve this problem is probably trial and error, but if you want the general solution with a more systematic approach:

    Let the roots of our quadratic ax^2 + bx + c be x_0\text{ and }x_1. We have x_0 + x_1 = -\frac ba = -5 and x_0x_1 = \frac ca = 6. This gives us the system of linear equations:

    \left\{\begin{array}{rcr}<br />
-b/a & = & -5\\<br />
c/a & = & 6<br />
\end{array}\right.

    \Rightarrow\left\{\begin{array}{rcr}<br />
 a & = & b/5\\<br />
a & = & c/6<br />
\end{array}\right.

    \Rightarrow\frac b5 = \frac c6\Rightarrow b = \frac{5c}6

    Now, let c be some arbitrary value t. Then we have as our solution:

    \left.\begin{array}{rcrcr}<br />
a & = & \frac c6 & = & \frac t6\smallskip\\<br />
b & = & \frac{5c}6 & = & \frac{5t}6\smallskip\\<br />
c & = & t<br />
\end{array}\right\}<br />
\Rightarrow(a,\;b,\;c) = \left(\frac t6,\;\frac{5t}6,\;t\right)

    Now just pick a (nonzero) value for t! For example, with t = 6 we have

    ax^2 + bx + c = x^2 + 5x + 6 = (x - 2)(x - 3)
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