The type of roots of a quadratic equation \(ax^2+bx+c=0\) can be determined by finding the value of discriminant, \(D=b^2-4ac\).
Determine the type of roots for the quadratic equation \(x^2+5x-6=0\).
Based on equation \(x^2+5x-6=0\),
\(a=1\), \(b=5\), \(c=-6\).
Use the formula for discriminant,
\(\begin{aligned} b^2-4ac&=5^2-4(1)(-6) \\ &=49 (\gt0) .\end{aligned}\)
Thus, the equation \(x^2+5x-6=0\) has two real and different roots.
The quadratic equation \(x^2+k+3=kx\), where \(k\) is a constant, has two equal real roots. Find the possible values of \(k\).
Arrange the equation in general form,
\(\begin{aligned} x^2+k+3&=kx\\ x^2-kx+k+3&=0. \end{aligned}\)
From the equation,
\(a=1\), \(b=-k\), \(c=k+3\).
Use the formula of discriminant,
\(\begin{aligned} b^2-4ac&=0\\ (-k)^2-4(1)(k+3)&=0\\ k^2-4k-12&=0\\ (k+2)(k-6)&=0 \end{aligned}\)
\(k=-2\) or \(k=6\).
Thus, the possible values for \(k\) are \(-2\) or \(6\).
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