Quadratic Formula Guide 2026: Discriminant, Roots and Factoring
Table of Contents
The Quadratic Formula
For ax^2 + bx + c = 0. Example: x^2 - 5x + 6 = 0 (a=1, b=-5, c=6).
x = (5 +/- sqrt(25 - 24)) / 2 = (5 +/- 1) / 2. Roots: x = 3, x = 2.
Check: (3)^2 - 5(3) + 6 = 9 - 15 + 6 = 0. Correct.
Missing terms are legal - just write their coefficients as zero. For 2x^2 - 7 = 0, read a = 2, b = 0, c = -7: x = (0 +/- sqrt(0 + 56))/4 = +/-sqrt(56)/4. Simplify: sqrt(56) = 2sqrt(14), so x = +/-sqrt(14)/2, about +/-1.87. Students who skip writing b = 0 often misalign the formula; writing the zero explicitly keeps every term in its lane.
The costliest exam error is plugging into the formula before rearranging. Given 3x^2 = 5x + 2, first move everything left: 3x^2 - 5x - 2 = 0. Then x = (5 +/- sqrt(25 + 24))/6 = (5 +/- 7)/6, giving x = 2 or x = -1/3. Check x = 2: 3(4) = 5(2) + 2, 12 = 12. The quadratic formula only works on equations equal to zero - standard form first, formula second.
The Discriminant Tells All
| D | Roots | Graph |
|---|---|---|
| D > 0 | Two real roots | Crosses x-axis twice |
| D = 0 | One repeated root | Touches x-axis once |
| D < 0 | Two complex roots | Never touches x-axis |
Example: x^2 + 2x + 5 = 0: D = 4 - 20 = -16. Complex roots: x = -1 +/- 2i.
The discriminant also predicts whether factoring is worth attempting. For x^2 - 7x + 10, D = 49 - 40 = 9 = 3^2 - a perfect square, so the roots are rational (5 and 2) and factoring will succeed. For x^2 - 5x + 3, D = 25 - 12 = 13, not a perfect square, so the roots are irrational and factoring over integers is hopeless - go straight to the formula. Compute D first and it tells you which method to use.
A zero discriminant has vivid real-world meaning. Suppose profit (in thousands) follows P = -2x^2 + 40x - 200 where x is hundreds of units. D = 1600 - 1600 = 0, so there is exactly one break-even quantity: x = -40/(2 x -2) = 10, or 1,000 units. The profit parabola just kisses the x-axis - produce 999 units and you lose money, 1,001 and you profit. A repeated root is the knife's edge between profit and loss.
Factoring vs Formula
- Factor first when a=1 and you can spot the pair: x^2 - 5x + 6 = (x-2)(x-3).
- Formula always works - use it when factoring isn't obvious.
- Completing the square reveals the vertex form: x^2 - 6x + 5 = (x-3)^2 - 4, vertex at (3, -4).
- Vertex shortcut: x-coordinate = -b/2a always.
The sum-product shortcut beats the formula whenever a = 1 and c has small factors. For x^2 - 8x + 15, hunt for two numbers with product 15 and sum -8: -3 and -5 fit, so the factorization is (x - 3)(x - 5). Listing c's factor pairs (1x15, 3x5) and checking their sums takes seconds. When no pair sums to b, abandon factoring gracefully and use the formula.
Special patterns deserve instant recognition. Difference of squares: x^2 - 64 = (x - 8)(x + 8). Perfect square trinomial: x^2 + 10x + 25 = (x + 5)^2. The pattern extends: 4x^2 - 9 = (2x - 3)(2x + 3), since both terms are squares. Spotting these forms turns a two-minute formula computation into a five-second rewrite.
Projectile Motion Application
Ball thrown upward: h(t) = -16t^2 + 48t + 6 (feet, seconds).
- Max height: t = -48/(2 x -16) = 1.5 s. h(1.5) = -36 + 72 + 6 = 42 ft.
- Hits ground: -16t^2 + 48t + 6 = 0. t = (-48 +/- sqrt(2304 + 384)) / -32 = (-48 +/- 51.8) / -32. Positive root: t = 3.12 s.
Projectiles pass each height twice, so expect two answers. With h(t) = -16t^2 + 48t + 6, when is the ball at 30 feet? Set -16t^2 + 48t + 6 = 30: -16t^2 + 48t - 24 = 0, or t^2 - 3t + 1.5 = 0. Then t = (3 +/- sqrt(9 - 6))/2 = (3 +/- sqrt(3))/2, about 0.63 s on the way up and 2.37 s on the way down. Two positive roots are not an error - they are the ascent and the descent.
Quadratics also solve every "maximize area" problem. With 100 feet of fence against a barn wall, let x be the depth: area A = x(100 - 2x) = -2x^2 + 100x. The vertex sits at x = -b/2a = -100/(2 x -2) = 25 feet, giving A = 25 x 50 = 1,250 sq ft. The same vertex formula that found the ball's peak height finds the biggest pen - optimization is just projectile math turned sideways.
Frequently Asked Questions
Always works for ax^2 + bx + c = 0. Try factoring first for speed; fall back to the formula when factoring isn't obvious.
b^2 - 4ac predicts the roots: positive = two real, zero = one repeated, negative = two complex.
x = -b/2a, then plug back for y. Or complete the square to vertex form a(x-h)^2 + k with vertex (h, k).
When D < 0, roots involve i (sqrt(-1)). They come in conjugate pairs and mean the parabola never crosses the x-axis.
Yes - projectile motion, profit optimization, area problems, and physics equations are full of quadratics.