Practise counting real roots

\(\Delta=0\), so there are equal roots.

\(\Delta=-23\), so there are equal roots.

\(\Delta=-23\), so there are no real roots.

\(\Delta=23\), so there are two distinct real roots.

\(\Delta=-41\), so there are no real roots.

\(\Delta=41\), so there are equal roots.

\(\Delta=41\), so there are two distinct real roots.

\(\Delta=0\), so there are equal roots.

\(\Delta=0\), so the roots are equal.

\(\Delta=144\), so there are two distinct real roots.

\(\Delta=-144\), so there are no real roots.

\(\Delta=0\), so there are no real roots.

\(\Delta=17\), so there are two distinct real roots.

\(\Delta=17\), so the roots are equal.

\(\Delta=0\), so the roots are equal.

\(\Delta=-17\), so there are no real roots.

\(\Delta=33\), so there are two distinct real roots.

\(\Delta=33\), so the roots are equal.

\(\Delta=0\), so the roots are equal.

\(\Delta=-33\), so there are no real roots.

\(\Delta=12\), so there are two distinct real roots.

\(\Delta=12\), so the roots are equal.

\(\Delta=0\), so the roots are equal.

\(\Delta=-12\), so there are no real roots.

\(-8\le a\le 8\)

\(-8<a<8\)

\(a=8\) or \(a=-8\)

\(a<-8\) or \(a>8\)

\(k=\dfrac16\)

\(k=-\dfrac14\)

\(k=-\dfrac16\)

\(k=\dfrac14\)

\(k\le\dfrac{25}{8}\)

\(k>\dfrac{25}{8}\)

\(k\ge\dfrac{25}{8}\)

\(k<\dfrac{25}{8}\) and \(k\neq0\)

\(k=4\) or \(k=-2\)

\(k=4\) only

\(k=2\) or \(k=-4\)

\(k=-2\) only