Use the discriminant b squared minus 4ac to determine whether a quadratic has two, one or no real roots and to solve related problems

What this dot point is asking

SEAB wants you to use the discriminant $b^2 - 4ac$ to decide, without solving, how many real roots a quadratic equation has, and to turn root conditions (two roots, equal roots, no real roots) into equations or inequalities for an unknown constant. The discriminant is the part of the quadratic formula under the square root, and its sign is what counts.

The answer

Where the discriminant comes from

The quadratic formula solves $ax^2 + bx + c = 0$:

The quantity $b^2 - 4ac$ under the root is the discriminant, written $\Delta$. Whether its square root is real, zero or imaginary decides the roots.

The three cases

When $\Delta = 0$ the curve just touches the $x$-axis; when $\Delta < 0$ it never reaches it.

Real roots means two or one

The phrase real roots usually means at least one, so $b^2 - 4ac \geq 0$. Read the wording carefully: distinct roots wants $\Delta > 0$, equal roots wants $\Delta = 0$.

Finding unknown constants

A typical problem gives a quadratic with an unknown $k$ and a condition on the roots. Translate the condition into a statement about $\Delta$, then solve the resulting equation or inequality for $k$.

The link to the graph

The discriminant describes how the parabola $y = ax^2 + bx + c$ meets the $x$-axis. Two real roots mean the curve cuts the axis twice; a repeated root means it just touches the axis at its vertex; no real roots means the curve stays entirely above or entirely below the axis. So a sketch and the discriminant tell the same story.

Discriminant of a quadratic in disguise

Conditions on roots often hide inside another problem, such as a line meeting a curve. After substituting to form a quadratic, the discriminant of that quadratic decides how many intersection points exist, which is why this idea recurs throughout coordinate geometry.

Examples in context

Example 1. A line tangent to a curve. Setting a line equal to a parabola and demanding one intersection gives a quadratic with $\Delta = 0$. Solving that condition finds the value of the gradient or intercept that makes the line a tangent, a recurring coordinate-geometry task.

Example 2. Guaranteeing a model has no solution. If a design requires that two paths never meet, the equation of their difference must have $\Delta < 0$, turning a geometric requirement into an algebraic inequality the discriminant supplies.

Try this

Q1. State the nature of the roots of $x^2 + 4x + 4 = 0$. [2 marks]

• Cue. $\Delta = 16 - 16 = 0$, so one repeated real root.

Q2. Find $k$ if $kx^2 + 4x + 1 = 0$ has equal roots. [3 marks]

• Cue. $16 - 4k = 0$, so $k = 4$.

Q3. For what values of $m$ does $x^2 + mx + 1 = 0$ have no real roots? [3 marks]

• Cue. $m^2 - 4 < 0$, so $-2 < m < 2$.