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Lesson · Grade 11 · Pre Calculus

The quadratic formula and the discriminant

  • Grade 11
  • Pre Calculus
  • 2 min 17 s
  • British Columbia
  • BC11-MA-76

The video solves 2x² + 3x − 2 = 0 with the quadratic formula: the discriminant is b² − 4ac = 9 + 16 = 25, so x = (−3 ± 5)/4 gives 1/2 or −2. It checks both roots, shows the common mistake of dividing only the square root by 2a, finds the roots as the x-intercepts of the parabola, then uses three graphs to show that a positive, zero or negative discriminant means two, one or no real roots, before ending on a one-line recap.

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In one sentence

The question this lesson answers

Let's solve 2 x squared plus 3 x minus 2 equals 0.

Compare with a x squared plus b x plus c equals 0: a is 2, b is 3 and c is negative 2.

Transcript

What the lesson says

Let's solve 2 x squared plus 3 x minus 2 equals 0.

Compare with a x squared plus b x plus c equals 0: a is 2, b is 3 and c is negative 2.

The quadratic formula: x equals negative b, plus or minus the square root of b squared minus 4 a c, over 2 a. The part under the root is the discriminant.

Here that is 3 squared minus 4 times 2 times negative 2, which is 9 plus 16, or 25.

Put the numbers in: negative 3, plus or minus the square root of 25, over 2 times 2.

That is negative 3 plus or minus 5, over 4.

Plus: negative 3 plus 5 is 2, and 2 over 4 is one half.

Minus: negative 3 minus 5 is negative 8, and negative 8 over 4 is negative 2.

Check both roots, or solutions: one half plus three halves minus 2 is 0, and so is 8 minus 6 minus 2.

A common mistake is dividing only the square root by 2 a. The whole top goes over 2 a.

On the graph, a curve called a parabola, the roots are the x-intercepts: where it crosses the x-axis.

The discriminant tells you how many real roots there are. Positive, like 25, means two.

Zero means one real root: x squared minus 6 x plus 9 gives 36 minus 36, so it touches the x-axis once.

Negative means no real roots: x squared plus 2 x plus 5 gives 4 minus 20, negative 16, so it never meets the x-axis.

Factoring, square roots and graphing solve some quadratics. The formula works for every one.

b squared minus 4 a c counts the real roots, and the formula finds them.

Curriculum

Where this fits

British Columbia, Grade 11

Pre Calculus 11, Quadratics: quadratic functions and equationsBC11-MA-76

Previous: Completing the Square: From Standard Form to Vertex Form

The outcomes and strands follow the British Columbia curriculum.

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