Convert ax² + bx + c into vertex form and solve step-by-step.
This calculator converts a quadratic expression in standard form ax² + bx + c into vertex form using the completing the square method. It also finds the vertex, axis of symmetry, discriminant, and real or complex roots instantly.
Automatically rewrites:
ax² + bx + c
into:
a(x − h)² + k
This makes it easy to identify the vertex and graph the parabola.
Perfect for algebra homework and exam preparation.
Choose between:
The calculator automatically detects whether roots are real or imaginary.
Enable “Show Step-by-Step” to see:
Great for understanding the completing the square method.
Step 1: Enter Your Quadratic Coefficients
Input the values of a, b, and c from your quadratic equation ax² + bx + c.
Step 2: Calculator Computes Key Values
Step 3: Equation Converted to Vertex Form
The calculator rewrites your equation as a(x − h)² + k for easy graphing and analysis.
Step 4: Roots Determined
Step 5: Optional Step-by-Step Guide
Check the “Show Step-by-Step” box to see exactly how the calculator:
This makes it easy to understand each step and learn the completing the square method interactively.
The completing the square method is a standard algebraic technique used to rewrite a quadratic equation ax² + bx + c into its vertex form a(x − h)² + k. This makes it easier to identify the vertex, axis of symmetry, and to solve for roots, whether they are real or complex.
a(x − h)² + k, where
h = −b / (2a), k = c − (b² / 4a)
In this formula:
This transformation allows you to easily:
Example Calculation: Convert x² + 6x + 5 into vertex form.
Step 1: h = −6 / 2 = −3
Step 2: k = 5 − (6² / 4) = 5 − 9 = −4
Step 3: Vertex form = (x + 3)² − 4
Here, the vertex is (−3, −4), the axis of symmetry is x = −3, and the parabola opens upward. This method can also be applied to equations with complex roots if the discriminant is negative.
If you enter:
a = 1, b = 6, c = 5
The calculator converts:
x² + 6x + 5
Into vertex form:
(x + 3)² − 4
Vertex: (−3, −4)
Roots: −1 and −5
Everything you need to know about converting quadratic equations into vertex form and finding roots step-by-step.
It converts a quadratic equation in standard form (ax² + bx + c) into vertex form using the completing the square method. It also calculates the vertex, axis of symmetry, discriminant, and roots.
Vertex form is written as a(x − h)² + k, where (h, k) represents the vertex of the parabola. This form makes graphing and analyzing quadratic functions easier.
If a = 0, the equation is no longer quadratic. The calculator will display an error because completing the square only applies to quadratic equations.
It uses the formula h = −b / 2a and k = c − (b² / 4a). The vertex is then displayed as (h, k).
The discriminant is calculated using D = b² − 4ac. It determines the nature of the roots: D > 0 → Two real roots, D = 0 → One real root, D < 0 → Complex roots.
Yes. If the discriminant is negative, select “Complex Roots” mode and the calculator will display the roots in the form a ± bi.
Yes. Simply check the “Show Step-by-Step” option to see how the equation is divided, adjusted, converted into a perfect square, and solved.
The precision dropdown lets you choose how many decimal places the results should display, from 1 to 10 decimal places.
Yes! The calculator is completely free and works instantly without registration or downloads.
Absolutely. The calculator is fully responsive and works smoothly on smartphones, tablets, laptops, and desktop computers.
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