How this calculator works
First factor a from the x^2 and x terms. Add and subtract the square of half the remaining x coefficient inside the brackets. One part forms a binomial square and the compensating constant becomes k. The vertex is (h, k). Solving the completed expression also reveals complex roots if the squared quantity must be negative.
Formula
ax^2 + bx + c = a(x + b/(2a))^2 + c - b^2/(4a). h = -b/(2a); k = c - b^2/(4a).
Worked example
For x^2 - 6x + 5, half of -6 is -3 and its square is 9. Add and subtract 9: (x - 3)^2 - 4. The vertex is (3, -4), and setting the expression to zero gives x = 1 or 5.
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Frequently asked questions
Why is a non-zero?
A square in x needs an x^2 term. With a = 0 the expression is linear.
Is the added term the same as k?
No. The added term completes the square inside the brackets. k includes the original constant and the compensation after multiplying by a.
Does this work when a is negative?
Yes. The parabola opens downward and the vertex gives its maximum rather than its minimum.