Use the discriminant b squared minus 4ac to determine the nature of the roots of a quadratic equation and apply it to find unknown constants

What this dot point is asking

SEAB wants you to use the discriminant of a quadratic equation, the quantity $b^2 - 4ac$, to decide how many real roots the equation has without actually solving it. You also need to run the idea backwards: given a condition such as "equal roots" or "no real roots", form an equation or inequality in an unknown constant and solve for it. The discriminant is the part of the quadratic formula that sits under the square-root sign, which is exactly why it controls the number of solutions.

The answer

Where the discriminant comes from

The roots of $ax^2 + bx + c = 0$ (with $a \ne 0$) are given by the quadratic formula:

The expression under the square root, $b^2 - 4ac$, is called the discriminant, often written as $\Delta$ (the Greek letter delta). Its sign decides what happens to the $\pm\sqrt{\ }$ part.

The three cases

• $b^2 - 4ac > 0$: the square root is a positive number, so $\pm$ gives two different values. The equation has two real and distinct roots, and the graph crosses the $x$-axis twice.
• $b^2 - 4ac = 0$: the square root is zero, so both signs give the same value. The equation has one repeated (equal) root, and the graph just touches the $x$-axis.
• $b^2 - 4ac < 0$: you would be taking the square root of a negative number, which has no real value. The equation has no real roots, and the graph does not meet the $x$-axis at all.

Running it backwards to find a constant

Many questions give you a quadratic containing an unknown letter and a condition on the roots. Translate the condition into the discriminant:

• "two distinct (or two real) roots" gives $b^2 - 4ac > 0$,
• "equal roots" (or "a repeated root", or "the line is a tangent") gives $b^2 - 4ac = 0$,
• "no real roots" gives $b^2 - 4ac < 0$.

Then substitute $a$, $b$ and $c$ in terms of the unknown and solve the resulting equation or inequality.

Identify a, b and c carefully

Always write the equation as $ax^2 + bx + c = 0$ first, including any signs. A negative coefficient is a frequent source of error, since $b^2$ makes the sign of $b$ vanish but the term $4ac$ keeps the signs of $a$ and $c$.

Examples in context

Example 1. Tangency to a curve. A straight line meets a parabola where their equations are equal. Setting them equal produces a quadratic; if the line is a tangent, that quadratic has equal roots, so $b^2 - 4ac = 0$. The discriminant turns a geometric tangency condition into one tidy algebraic equation, which is why it appears in coordinate-geometry questions too.

Example 2. Guaranteeing real solutions. An engineer models a system with $x^2 - 6x + c = 0$ and needs the model to have real solutions for it to be physically meaningful. The condition is $b^2 - 4ac \ge 0$, that is $36 - 4c \ge 0$, so $c \le 9$. The discriminant sets the exact range of the constant for which real answers exist.

Try this

Q1. Find the discriminant of $3x^2 + 2x - 1 = 0$ and state the nature of its roots. [2 marks]

• Cue. $b^2 - 4ac = 2^2 - 4(3)(-1) = 4 + 12 = 16 > 0$, so two real and distinct roots.

Q2. The equation $x^2 - 8x + c = 0$ has equal roots. Find $c$. [2 marks]

• Cue. Set $b^2 - 4ac = 0$: $64 - 4c = 0$, so $c = 16$.

Q3. Show that $x^2 + x + 1 = 0$ has no real roots. [2 marks]

• Cue. $b^2 - 4ac = 1 - 4 = -3 < 0$, so there are no real roots.