Quadratics — FAQ

They are closely related concepts. 'Roots' or 'solutions' refer to the algebraic values of $x$ that solve the equation $ax^2+bx+c=0$. 'x-intercepts' are the geometric points where the graph of the function $y=ax^2+bx+c$ crosses the x-axis. The x-coordinates of these intercepts are the real roots of the equation.

Always try to factorise first, as it's faster. If you can't spot the factors within about 20-30 seconds, or if the question asks for an answer to a specific number of decimal places or significant figures, it's a strong hint that you should use the quadratic formula.

If you have $ax^2+bx+c$, you must first factor out the coefficient 'a' from the first two terms, giving $a(x^2 + (b/a)x) + c$. Then, you complete the square inside the bracket as usual and multiply out at the end. See the second worked example for a step-by-step guide.

Graphically, 'no real roots' means the parabola does not touch or cross the x-axis at all. If the coefficient 'a' is positive, the entire curve will be above the x-axis. If 'a' is negative, the entire curve will be below the x-axis.

Only when $a > 0$. If $a < 0$ the parabola is n-shaped and the vertex is the highest point (a maximum). The sign of $a$ decides minimum versus maximum every time — try dragging the vertex above the x-axis with a negative $a$ in the explorer to see it flip.

Narrower. Increasing $|a|$ stretches the curve vertically so it rises more steeply and looks narrower; values of $|a|$ between 0 and 1 make it wider and flatter, and a negative $a$ flips it upside-down.