1) For what values of \(p\) does \(4x^2-3x+p=0\) have at least one real root?
2) Find the values of \(k\) so that \(3x^2+6x-k=0\) has exactly two real roots.
3) For what values of \(p\) does \(px^2 + 4\sqrt{3x} + 7 - p = 0\) have exactly one solution?
4) For what values of \(n\) does \({\begin{cases} y=x^2-4x-n\\y=3x+2x^2\end{cases}}\) have no real solutions?
5) Use the discriminant to determine the number of real roots for \(\large\frac{\sqrt{5}}{2}\normalsize{x^2}-\large\frac{\sqrt{2}}{4}\normalsize{x}+\large\frac{1}{16}\normalsize=0\).
6) Use the discriminant to determine the number of real roots for \(7x^2+2=4\sqrt{5}x\).
7) Let \( f(x)=x^2 +x \). Does \(4f(a)=f(b) \) have integer solutions \(a \) and \(b \)? Prove your claim. Hint: discriminant
Solution Bank
2) Find the values of \(k\) so that \(3x^2+6x-k=0\) has exactly two real roots.
3) For what values of \(p\) does \(px^2 + 4\sqrt{3x} + 7 - p = 0\) have exactly one solution?
4) For what values of \(n\) does \({\begin{cases} y=x^2-4x-n\\y=3x+2x^2\end{cases}}\) have no real solutions?
5) Use the discriminant to determine the number of real roots for \(\large\frac{\sqrt{5}}{2}\normalsize{x^2}-\large\frac{\sqrt{2}}{4}\normalsize{x}+\large\frac{1}{16}\normalsize=0\).
6) Use the discriminant to determine the number of real roots for \(7x^2+2=4\sqrt{5}x\).
7) Let \( f(x)=x^2 +x \). Does \(4f(a)=f(b) \) have integer solutions \(a \) and \(b \)? Prove your claim. Hint: discriminant
Solution Bank