The quadratic formula solves ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) ÷ 2a. For x² − 5x + 6 = 0, the discriminant is 25 − 24 = 1, so x = (5 ± 1) ÷ 2: x = 2 and x = 3. A negative discriminant means there are no real roots, only complex ones.
My Construction Calculator
Quadratic Formula Calculator
Inputs
Results update as you type.
Diagram updates as you type.
Results
Check the working before you order.
Roots
x = 2 and x = 3
Discriminant (b² − 4ac)
1
Type of roots
Two real roots
Vertex (turning point)
(2.5, −0.25)
Axis of symmetry
x = 2.5
y-intercept
(0, 6)
Price This optional: add your own prices
Your prices are remembered on this device only.
Send Order WhatsApp, SMS or email to your supplier
x = (5 ± √1) ÷ 2 = (5 ± 1) ÷ 2, so x = 2 and x = 3
Vertex at x = −b ÷ 2a = 2.5, y = −0.25
Scan to open this calculationThe code links to this calculator with your inputs, so anyone can scan it with a phone camera and see the same result. Save it for a report, a site board or WhatsApp.Test copy: this code opens the live site, myconstructioncalculator.com, so it works on a phone once the site is launched.
Results are estimates for planning and material ordering only. They are not a substitute for structural design or approval by a qualified engineer. Read the full disclaimer.
Scan to open this calculation with the same inputs.
Generated with myconstructioncalculator.com. Estimates for planning and ordering only; verify against drawings and specifications.
Formula
ax² + bx + c = 0x = (−b ± √(b² − 4ac)) ÷ 2aDiscriminant D = b² − 4acVertex: x = −b ÷ 2a, y = c − b² ÷ 4a
Worked Example
A slab is 3 m longer than it is wide and has an area of 40 m². Find the width w:
w × (w + 3) = 40, so w² + 3w − 40 = 0 (a = 1, b = 3, c = −40)
D = 3² − 4 × 1 × (−40) = 169, √169 = 13
w = (−3 ± 13) ÷ 2 = 5 or −8; a width cannot be negative, so the slab is 5 × 8 m
Examples
Equation
Discriminant
Roots
x² − 5x + 6 = 0
1
2 and 3
2x² + 3x − 2 = 0
25
−2 and 0.5
x² − 4x + 4 = 0
0
2 (repeated)
x² − 2 = 0
8
±1.4142
x² + 2x + 5 = 0
−16
−1 ± 2i
Frequently Asked Questions
What is the quadratic formula?
x = (−b ± √(b² − 4ac)) ÷ 2a, for any equation ax² + bx + c = 0 where a is not 0. The ± gives the two roots.
What does the discriminant tell you?
b² − 4ac decides the roots. Positive: two different real roots. Zero: one repeated root. Negative: no real roots, but two complex roots.
What if a is 0?
Then there is no x² term and the equation is linear: bx + c = 0, so x = −c ÷ b. The quadratic formula does not apply.
Where are quadratic equations used in construction?
Whenever an area or a curve depends on an unknown length. A slab 3 m longer than it is wide with an area of 40 m² gives w² + 3w − 40 = 0, so w = 5 m and the slab is 5 × 8 m.
What is the vertex of a parabola?
Its turning point, at x = −b ÷ 2a. For x² − 5x + 6 it is at (2.5, −0.25), the lowest point of the curve.
Embed This Calculator on Your Website
Free for any website: contractors, suppliers, blogs and schools. Paste this code where you want the calculator to appear. It works in WordPress (Custom HTML block), Wix, Squarespace and plain HTML.
