Short answer: for ax² + bx + c = 0, the roots add up to -b/a and multiply to c/a. The vertex is at x = -b/(2a). Many questions ask for something you can get from these directly, without finding the roots.
The three facts
For the equation ax² + bx + c = 0 with roots r and s:
- Sum of roots: r + s = -b/a
- Product of roots: r · s = c/a
- Vertex (axis of symmetry): x = -b/(2a)
Example
Question: The equation 2x² - 12x + 7 = 0 has solutions r and s. What is the value of r² + s²?
Slow route: use the quadratic formula, get two messy numbers with a square root, square each and add. Very error-prone.
Shortcut:
- Sum: r + s = -(-12)/2 = 6
- Product: r · s = 7/2
- Use the identity r² + s² = (r + s)² - 2rs
- So r² + s² = 6² - 2(7/2) = 36 - 7 = 29
No roots needed.
Other common questions this solves
- “What is the sum of the solutions?” Read it from -b/a.
- “What is the minimum or maximum value?” Find the vertex x with -b/(2a), then put it back in the function.
- “How many solutions does it have?” Check the discriminant b² - 4ac. Positive means two, zero means one, negative means none.
Combine it with Desmos
If you forget a formula, graph the parabola. The vertex and x-intercepts show on screen. Knowing both methods lets you check one with the other on a question you are not sure about.
Get the full method
Quadratics and equivalent expressions are a full module in the course. Book a free call to see where your own points are going.