Library / Foundations: Maths / FMA-19

Foundations: Maths

Quadratic formula

Larger root of ax² + bx + c = 0.

Ref: FMA-19
Rev: v1
Field: Foundations: Maths
Result: x₊

x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}

For ax² + bx + c = 0 with a non-zero. A negative discriminant b² − 4ac gives no real roots. The calculator shows the larger root, with a > 0, b < 0 and c > 0.

The quadratic formula gives the values of x where a parabola crosses the axis, using only the three coefficients.

Calculator

Solve for

x₊Larger rootComputed

aCoefficient a

0.54

bCoefficient b

−12−2

cCoefficient c

0.510

Smaller root2
Discriminant b² − 4ac1

What drives the result

Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.

Larger root () current value 3

bCoefficient bKey driver

11.5

aCoefficient a

8.61

cCoefficient c

2.724.9

The roots add up to −b/a and multiply to c/a, a useful check.
When the discriminant is zero there is one repeated root.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing ()	Local step	Change in result
b Coefficient b	−12 to −2	3	11.48	8.477	+10%	+33.3 %
a Coefficient a	0.5 to 4	3	8.606	5.606	+10%	—
c Coefficient c	0.5 to 10	2.724	4.898	2.174	+10%	—

Sweep of Coefficient c

Coefficient c	Larger root
0.5	4.898
1.688	4.636
2.875	4.337
4.063	3.979
5.25	3.5
6.438	—
7.625	—
8.813	—
10	—

Sweep one variable

Vary

Larger root ()Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

x₊

Larger root

Result ·

What the equation gives you. Everything else feeds it.

Coefficient a

· typical 0.5 to 4

Coefficient of x². Larger a squeezes the roots towards zero.

Coefficient b

· typical −12 to −2

Coefficient of x (negative here). Larger |b| spreads the roots apart.

Coefficient c

· typical 0.5 to 10

Constant term. Larger c brings the roots together until they vanish.

Worked example with the values above

Discriminant: b² − 4ac = 25 − 24 = 1.
Larger root: (−b + √D)/(2a) = (5 + 1) / 2.
x = 3.

Assumptions and limits

a is non-zero.
Real coefficients; only real roots are reported.

Common mistakes

Dropping the sign of b.
Forgetting to divide the whole numerator by 2a.

Statistics and uncertainty

Near a zero discriminant the square root amplifies small coefficient errors, so the roots are poorly determined when they are close together.

Where it is used

Time of flight and landing points from a quadratic motion model.
Break-even and design-limit equations.

Maths you need

Square roots: √D
Substitution: with negative numbers

Related formulas

Practise this formula