Completing the square calculator
x² + 6x + 5 becomes (x + 3)² - 4 in the completing the square calculator, so x = -1 or x = -5 and the vertex is (-3, -4).
Roots
- Exact form
- x = -1 or x = -5
- Discriminant Positive, so two distinct real roots, and a perfect square, so both are rational.
- 16
- The formula applied
- x = (-6 +/- sqrt(16)) / 2
- Vertex A minimum, since a is positive and the parabola opens upward.
- (-3, -4)
- Axis of symmetry
- x = -3
- Factored form
- (x + 1)(x + 5) = 0
- Y-intercept
- (0, 5)
- Sum and product of the roots
- -6 and 5
- Completed square
- (x + 3)^2 - 4
How the Quadratic Formula Calculator works
Completing the square rewrites ax² + bx + c in vertex form, a(x - h)² + k. The completing the square formula is h = -b ÷ 2a and k = c - b² ÷ 4a.
By hand, for x² + 6x + 5: take half the coefficient of x, 6 ÷ 2 = 3, and square it, 9. Add it and take it away again: x² + 6x + 9 - 9 + 5. The first three terms are now a perfect square, so the expression is (x + 3)² - 4.
To solve (x + 3)² - 4 = 0, move the 4 across and take square roots: x + 3 = ±2, so x = -1 or x = -5.
When a is not 1, factor it out of the x terms first. 2x² + 8x + 3 = 2(x² + 4x) + 3 = 2(x + 2)² - 8 + 3 = 2(x + 2)² - 5, with its vertex at (-2, -5).
Done with letters instead of numbers, the same steps derive the quadratic formula: a(x + b/2a)² = b²/4a - c, which rearranges to x = (-b ± √(b² - 4ac)) ÷ 2a.
Vertex form also draws the graph. The parabola's lowest point, or its highest when a is negative, is at (h, k), and the axis of symmetry is the line x = h.
Quadratic Formula Calculator FAQ
What is the vertex form of x² - 4x + 1?
(x - 2)² - 3. Half of -4 is -2, its square is 4, and 1 - 4 = -3. The vertex is (2, -3).
Why add and subtract the same number?
Adding (b/2)² makes a perfect square, and subtracting it again keeps the expression equal to the original. Together they add zero.
When is completing the square easier than the quadratic formula?
When a is 1 and b is even, because halving b leaves whole numbers. It is also the one of the two that gives the vertex directly.