The number under the root

The idea

For ax² + bx + c = 0, the expression b² – 4ac is called the discriminant, because it determines whether the equation has real roots or not.

What could go wrong?

The formula x = (−b ± √(b² − 4ac)) / 2a has a square root in it. Take 2x² − 4x + 3 = 0 and look at the part under the root.

What could go wrong with the formula?

• Nothing: it works for every equation
• The part under the root might be negative
• The answer might be too big

Watch the formula run and jam

Step a, b and c. The machine works out D = b² − 4ac and tells you if the formula can run. Make it run, and make it jam.

The discriminant

For ax² + bx + c = 0, the expression b² − 4ac is called the discriminant, D. It decides whether the equation has real roots or not.

If D ≥ 0, then √D is a real number and the formula gives real roots. If D < 0, no real number multiplied by itself gives D, so there are no real roots.

The book's example: 2x² − 4x + 3 = 0 has a = 2, b = −4, c = 3, so D = (−4)² − 4 × 2 × 3 = 16 − 24 = −8. It is negative, so there are no real roots. The name fits: D discriminates (tells apart) the equations with real roots from those without.

Notes

For ax² + bx + c = 0, the discriminant is D = b² − 4ac. It decides whether there are real roots: D ≥ 0 the formula runs; D < 0 there are no real roots.

Check yourself

Find the discriminant of x² + 7x − 60 = 0 (the pole in the park).

Answer: 289

b² − 4ac = 7² − 4 × 1 × (−60) = 49 + 240 = 289. It is positive, so the equation has real roots.

Find the discriminant of x² − 3x + 5 = 0.

Answer: -11

b² − 4ac = (−3)² − 4 × 1 × 5 = 9 − 20 = −11.

The discriminant of an equation is −11. What does that tell you?

Why is b² − 4ac called the discriminant?

• It has no real roots: √(−11) is not a real number — correct. Yes! A negative D means no real roots.
• It has two real roots. The formula needs √D. A negative D has no real square root.
• It is not a quadratic equation. A negative D does not stop it being quadratic. It only means there are no real roots.
• It discriminates: it tells whether real roots exist or not — correct. Yes! Its sign decides it.
• It is the difference of two squares. That is not the reason. The name comes from what it does: it tells the equations apart.
• It is always negative. It can be negative, zero or positive. Its value tells whether the roots are real.