Discriminant

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Discriminant: meaning, method and solved example (Algebra 1) – Maths PDF

In short: Discriminant is the quantity b² − 4ac built from a quadratic’s coefficients; whether it is positive, zero or negative predicts how many real solutions exist.

Algebra 1AlgebraA-REI.B.4b

What is Discriminant?

Before doing all the work of the quadratic formula, glance at the number sitting under its radical: that number, \(b^2 – 4ac\), is the discriminant, and it forecasts the kind of answers without computing them.

For \(ax^2 + bx + c = 0\), a positive result signals two distinct real answers, zero signals a single repeated answer, and a negative result signals none, since real numbers have no square root of a negative. Graphically that means two crossings, one touch, or a curve that never reaches the axis.

How to spot it

  • A question asks how many real solutions or x-intercepts exist, but not what they are.
  • A parameter k must be chosen so that an equation has exactly one answer.
  • The expression \(b^2 – 4ac\) shows up under a radical in a solution.
  • A model asks whether a curve ever reaches a stated height.

The method

  1. Set the equation equal to zero and read a, b and c with their signs.
  2. Compute \(b^2\), then subtract \(4ac\), watching negative signs.
  3. Sort the result: positive gives two real solutions, zero gives one, negative gives none.
  4. If real answers are wanted, carry the same value into the quadratic formula.
positive2 real solutionszero1 real solutionnegativeno real solutionssign of b² − 4ac
The sign of b² − 4ac decides how many times the parabola meets the x-axis.

Discriminant: solved example

Problem

A basketball’s height is modeled by \(h = -0.5x^2 + 3x – 6\) feet, with x the horizontal distance. Without solving fully, decide whether the ball ever reaches a height of 3 feet.

Solution

Set \(-0.5x^2 + 3x – 6 = 3\), which becomes \(-0.5x^2 + 3x – 9 = 0\). Multiply by \(-2\): \(x^2 – 6x + 18 = 0\), so \(a = 1\), \(b = -6\), \(c = 18\).

The discriminant is \(36 – 72 = -36\), which is negative. The ball never reaches 3 feet; its highest point stays below that.

Try it

Question 1

How many real solutions does \(2x^2 – 5x + 4 = 0\) have?

Show the answer

\(25 – 32 = -7\) is negative, so none.

Question 2

For which k does \(x^2 + 6x + k = 0\) have exactly one solution?

Show the answer

Need \(36 – 4k = 0\), so \(k = 9\).

Discriminant vs. Quadratic Formula

The discriminant is only the piece under the radical, one number that sorts the cases. The quadratic formula is the whole recipe that produces the solutions. Computing the first costs seconds and can save you from a long calculation that would end with a negative square root.

Quadratic Formula

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