Completing the Square Calculator
Convert quadratic equations ax^2 + bx + c = 0 into vertex form y = a(x - h)^2 + k by completing the square step-by-step.
Completing the Square Engine
Enter coefficients a, b, and constant c.
Vertex Form Output
Quick Summary
Our completing the square calculator vertex form steps tool converts standard quadratic equations $ax^2 + bx + c = 0$ into vertex form $y = a(x - h)^2 + k$. It calculates the square-completion term $\left(\b / 2a\right)^2$, finds vertex coordinates $(h, k)$, and solves for roots $x$.
How It Works: Completing the Square Method
Completing the square is an algebraic technique used to rewrite quadratics as perfect squares:
1. **Isolate Constant:** Move constant $c$ to right side: $x^2 + \left(\b / a\right)x = -\c / a$.
2. **Half the Linear Term:** Take half of linear coefficient: $\b / 2a$.
3. **Square and Add:** Square that value $\left(\b / 2a\right)^2$ and add to both sides.
4. **Factor Perfect Square:** Factor left side into $\left(x + \b / 2a\right)^2$.
5. **Solve for x:** Take square root of both sides and solve $x = -\b / 2a \pm \sqrt{\left(\b / 2a\right)^2 - \c / a}$.
Formula Explanation
Your square completion calculations follow vertex transformation formulas:
Step-by-Step Worked Example
Here is a detailed 5-step breakdown for completing the square on x² - 6x + 5 = 0:
- Step 1 (Isolate Variables): $x^2 - 6x = -5$.
- Step 2 (Half Linear Term & Square): Half of $-6$ is $-3$. Square it: $(-3)^2 = \mathbf{9}$.
- Step 3 (Add 9 to Both Sides): $x^2 - 6x + 9 = -5 + 9 \implies \mathbf{x^2 - 6x + 9 = 4}$.
- Step 4 (Factor Perfect Square): $(x - 3)^2 = 4$.
- Step 5 (Take Square Root & Solve): $x - 3 = \pm \sqrt{4} = \pm 2 \implies x = 3 \pm 2$, delivering **x₁ = 5, x2 = 1**!
Calculation Examples: Real-World Scenario Comparison
Compare standard equations, completed square terms, vertex coordinates, and solved roots:
| Standard Quadratic | Term Added (b/2a)² | Completed Square Eq | Vertex Form | Parabola Vertex (h, k) | Solved Roots |
|---|---|---|---|---|---|
| x² - 6x + 5 = 0 | 9 | (x - 3)² = 4 | y = (x - 3)² - 4 | (3, -4) | x&sb1; = 5, x&sb2; = 1 |
| x² + 8x + 12 = 0 | 16 | (x + 4)² = 4 | y = (x + 4)² - 4 | (-4, -4) | x&sb1; = -2, x&sb2; = -6 |
| 2x² - 12x + 10 = 0 | 9 | 2(x - 3)² = 8 | y = 2(x - 3)² - 8 | (3, -8) | x&sb1; = 5, x&sb2; = 1 |
| x² - 4x + 9 = 0 | 4 | (x - 2)² = -5 | y = (x - 2)² + 5 | (2, 5) | 2 ± √5 i |
Benefits of Using the Completing the Square Calculator
Utilizing this calculator provides essential conic section graphing, calculus integration, and derivation advantages:
- Direct Vertex Form Conversion: Instantly converts standard $ax^2 + bx + c$ into vertex form $a(x - h)^2 + k$.
- Derivation of Quadratic Formula: Shows the exact algebraic steps that prove the quadratic formula.
- Conic Section Equations (Circles, Ellipses, Hyperbolas): Converts general conic equations $(x - h)^2 + (y - k)^2 = r^2$ into standard graphing form.
- Calculus Trigonometric Substitution ($\int \dx / x^2 + bx + c$): Completes the square under integrals for inverse tangent substitutions.
Frequently Asked Questions (FAQ)
What does completing the square mean?
Completing the square adds a specific constant term (b/2a)^2 to a quadratic expression to convert it into a perfect square binomial (x - h)^2.
How do you find the term to add to complete the square?
Take half of the coefficient of x (b / 2a) and square it: (b / 2a)^2.
What is vertex form of a quadratic equation?
Vertex form is y = a(x - h)^2 + k, where (h, k) is the vertex point of the parabola.
How do you complete the square for x^2 - 6x + 5 = 0?
x^2 - 6x = -5. Half of -6 is -3, squared is 9. Add 9: x^2 - 6x + 9 = 4 ⇒ (x - 3)^2 = 4 ⇒ x = 3 ± 2 ⇒ x1=5, x2=1.
What if coefficient 'a' is not 1 (e.g. 2x^2 - 12x + 10 = 0)?
Divide the entire equation by a (or factor a out from variable terms) first: 2(x^2 - 6x) = -10 ⇒ 2(x - 3)^2 = 8.
How does completing the square relate to the quadratic formula?
The quadratic formula x = (-b ± √(b^2 - 4ac)) / (2a) is derived by completing the square on the general equation ax^2 + bx + c = 0.
Why is vertex form useful?
Vertex form clearly displays the maximum or minimum point (h, k) of a parabola without requiring graphing.
Can completing the square solve complex roots?
YES! If (x - h)^2 = negative number, taking the square root yields imaginary solutions ± i√N.
What is the axis of symmetry in vertex form?
The vertical line x = h.
How is completing the square used for circles?
It converts x^2 + y^2 + Dx + Ey + F = 0 into center-radius form (x - h)^2 + (y - k)^2 = r^2.