The Discriminant and Nature of Roots

The Discriminant and Nature of Roots

The discriminant $\Delta=b^2-4ac$ tells the nature of roots before you solve. If $\Delta>0$, there are two distinct real roots; if $\Delta=0$, one repeated real root; if $\Delta<0$, no real roots. Graphically, this describes whether a parabola crosses, touches or misses the $x$-axis.

Worked reasoning

For $x^2+2x+5=0$, $\Delta=2^2-4(1)(5)=4-20=-16$. It has no real roots, so its graph does not meet the $x$-axis.

Exam method

1. Identify signed coefficients. 2. Calculate $b^2-4ac$. 3. State the root type in words, and connect it to intercepts if asked.

Predict before solving

The discriminant is an efficient planning tool. Calculate it before choosing a factorisation or formula route. A perfect-square positive discriminant usually gives rational roots; a positive non-square gives irrational roots; zero gives a tangent point; negative rules out real x-intercepts. In a graph question, write both the algebraic and graphical conclusion. Connecting ‘$\Delta<0$’ to ‘no intersection with the x-axis’ demonstrates conceptual understanding rather than formula recall.

One-minute retrieval: The Discriminant and Nature of Roots

Close the worked solution. From memory, state its central rule, name one condition that makes it valid, and reconstruct one check that would catch a typical exam error.