- Determine the nature of the roots of \( \displaystyle 3x^2 + 2x - 6 = 0 \)
Solution
2 real irrational roots
- Determine the nature of the roots of \( \displaystyle x^2 - 2\sqrt{5}x + 5 = 0 \)
Solution
one real rational (double) root
- Find the values of \(b\) and \(c\) if the equation \( \displaystyle 4x^2 + bx + c = 0 \) has as its roots \( \displaystyle \frac{1 + \sqrt{2}}{2},\) and \( \displaystyle \frac{1 - \sqrt{2}}{2}.\)
Solution
\( \displaystyle b = -4; c = -1 \)
- Find a quadratic equation whose roots are \( \displaystyle 1 + i\sqrt{3} \) and \( \displaystyle 1 - i\sqrt{3} \)
Solution
\( \displaystyle x^2 - 2x + 4 = 0 \) or any scalar multiple thereof. (That is, I'd accept anything in the form \( \displaystyle a(x^2 - 2x + 4) = 0 \) for any value of \(a.\)
- Determine the value(s) of \(k\) for which the given equation will have exactly one real root: \( \displaystyle 5x^2 - kx + 8 = 0 \)
Solution
\( \displaystyle k = \pm 4\sqrt{10} \)
- Determine the value(s) of \(k\) for which the given equation will have imaginary roots. \( \displaystyle x^2 + 4(k + 1)x + 4k^2 = 0 \)
Solution
\( \displaystyle k < -\frac{1}{2} \)
- Determine the value(s) of \(k\) so that one root of \(16x^2 - 9x + k = 0\) is three times the other.
Solution
\( \displaystyle \frac{243}{256} \)