Equations
The Quadratic Formula and the Discriminant
The formula that solves every quadratic, what b2 - 4ac tells you before you compute anything, and the three cases it distinguishes.
What this page covers
- Apply the quadratic formula correctly, including signs
- Compute the discriminant and predict the nature of the roots
- Distinguish two real, one repeated and two complex roots
- Use the sum and product of roots as a check
The formula
It is not a separate technique. It is completing the square carried out once on the general case, so that it never has to be done again. The derivation is on Completing the Square.
- Rearrange to = 0. The formula assumes it.
- Identify a, b, c with their signs. Write them down explicitly; most errors enter here.
- Compute b2 - 4ac first. It tells you what kind of answer to expect before any square roots are taken.
- Substitute and simplify. Keep the whole numerator over 2a.
- Check with the sum and product of the roots.
b2 is positive even when b is negative. For b = -5, b2 = 25, not -25. Writing b in brackets as (-5)2 prevents this.
The discriminant
Δ = b2 - 4ac, the quantity under the radical. The source names it explicitly on page 5.
| b2 - 4ac | Roots | Graph | Source example |
|---|---|---|---|
| Positive | Two distinct real roots | Crosses the x-axis twice | 2x2 + 3x - 2 = 0 |
| Zero | One repeated real root | Touches the axis at the vertex | x2 + 4x + 4 = 0 |
| Negative | Two complex conjugate roots | Never meets the axis | x2 + 4x + 5 = 0 |
A fourth distinction is useful when integer factors are wanted: if the discriminant is a perfect square, the roots are rational and the quadratic factors over the integers. If it is positive but not a perfect square, the roots are irrational and arrive as a conjugate surd pair.
The three cases, worked
Positive discriminant — 2x2 + 3x - 2 = 0
a = 2, b = 3, c = -2.
Note -4ac = -4(2)(-2) = +16. Two negatives make the term positive, which is where sign errors cluster.
Sum of roots should be -ba = -32: 12 - 2 = -32. Product should be ca = -1: 12 × -2 = -1. Both hold.
Zero discriminant — x2 + 4x + 4 = 0
One root, repeated. The source confirms it by factoring: (x + 2)2 = 0, so x = -2 twice. The parabola touches the axis rather than crossing it.
Negative discriminant — x2 + 4x + 5 = 0
The source first writes 'NO REAL SOLUTION' and then, on the following page, gives the complex answer. Both are correct statements about different number systems.
Sum = -4 = -ba. Product = (-2 + i)(-2 - i) = 4 - i2 = 5 = ca. Both hold.
A source error worth recording
Week 6, page 4 solves 2x2 - 5x + 3 = 0 and writes the formula line as 25 ± √14 = 264, 244. The 25 has been carried down from underneath the radical. The correct working is x = 5 ± √25 - 244 = 5 ± 14 = 32 or 1. Substituting confirms: 2(94) - 152 + 3 = 0 and 2 - 5 + 3 = 0.
The error is instructive. Under the radical sits b2 - 4ac; outside it sits -b. Confusing the two is easy in handwriting, and the sum-of-roots check catches it at once: 264 + 244 = 12.5, whereas -ba = 52 = 2.5.
Sum and product of the roots
These follow from expanding a(x - α)(x - β) and comparing coefficients. As a check they are excellent: two quick arithmetic tests that use both roots at once and are independent of the method used to find them.
| Equation | Roots | Sum vs -ba | Product vs ca |
|---|---|---|---|
| 2x2 + 3x - 2 | 12, -2 | -32 ✓ | -1 ✓ |
| x2 + 4x + 4 | -2, -2 | -4 ✓ | 4 ✓ |
| x2 + 4x + 5 | -2 ± i | -4 ✓ | 5 ✓ |
Common mistakes
| Mistake | Correct | Prevention |
|---|---|---|
| b2 taken as negative when b < 0 | Always positive | Write (-5)2 in brackets |
| Sign lost in -4ac when c < 0 | -4(2)(-2) = +16 | Bracket every substitution |
| Only the radical divided by 2a | The whole numerator is divided | Keep the fraction bar long |
| Applying the formula when a = 0 | Not a quadratic | Check a ≠ 0 first |
| Concluding 'no solution' without saying over which system | No real solution | State the number system |
| Carrying -b under the radical | -b is outside | Check with the sum of roots |
Frequently asked questions
What does the discriminant actually measure?
How far the two roots sit either side of the vertex. Zero discriminant means they coincide at the vertex; positive means they separate; negative means the parabola never reaches the axis and the separation is imaginary.
Why is it -b and not b?
Because the vertex is at x = -b2a, and the roots sit symmetrically about it. The sign comes out of completing the square, where the substitution is x + b2a.
Can I use the formula when the quadratic factors easily?
Yes, and it will give the right answer. Factoring is usually faster when the numbers are small, but the formula never fails and never requires a search.
What if a = 0?
Then the equation is not quadratic and the formula divides by zero. Check that a ≠ 0 before applying it — a linear equation needs a linear method.
Source. Handwritten teaching notes, Week 3, pages 5-7, and Week 6, page 4.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates, reorganises and verifies the mathematics of the supplied teaching notes; it is not a reproduction of them. Numerical values taken from the notes are identified as source examples and are not presented as engineering standards.