Vertex form to standard form calculator

Enter the leading coefficient a and the vertex coordinates h and k. The calculator applies the distributive property, combines the constant terms, and reports a, b, and c directly. It is useful for checking algebra, preparing an equation for the quadratic formula, or moving between two common descriptions of the same parabola without changing the graph.

Reading vertex form correctly

A quadratic in vertex form is written as a(x − h)² + k. The number a controls the direction and vertical scale of the parabola, while the point (h, k) is its vertex. Enter h with the sign it has as a coordinate, not merely the sign printed inside the parentheses. For example, (x − 3)² uses h = 3, but (x + 3)² is the same as (x − (−3))² and therefore uses h = −3. The leading coefficient a cannot be zero, because zero would remove the squared term and the expression would no longer be quadratic. Both h and k may be zero, negative, positive, or decimal values. Once those three values are supplied, the conversion is completely determined: there is exactly one standard-form polynomial representing the same function. The calculator returns only finite numeric coefficients, so the result can be copied directly into later algebraic work or checked against a manual expansion.

How the expansion produces b and c

The conversion starts with the identity (x − h)² = x² − 2hx + h². Multiplying every term by a gives ax² − 2ahx + ah², and adding k produces ax² − 2ahx + ah² + k. Comparing that expression with ax² + bx + c shows that the leading coefficient stays a, the linear coefficient is b = −2ah, and the constant coefficient is c = ah² + k. Consider 2(x − 3)² − 5: expanding the square gives 2(x² − 6x + 9) − 5, which becomes 2x² − 12x + 18 − 5 and finally 2x² − 12x + 13. The calculator performs these same identities directly rather than manipulating a formatted equation string. That makes signs explicit and avoids ambiguity around superscripts, multiplication symbols, or alternative keyboard notation. It also means decimals are treated as ordinary numeric coefficients and follow standard JavaScript floating-point arithmetic.

Checking and using the standard-form result

A quick way to check the conversion is to substitute x = h into the standard-form result. Because h is the vertex’s horizontal coordinate, ah² + bh + c should equal k. You can also compare values from the original and expanded expressions at any convenient x; matching values confirm that the two forms describe the same parabola. Standard form is especially useful when you need to identify the constant term immediately, apply the quadratic formula, compute the discriminant b² − 4ac, add or subtract polynomials, or compare coefficients across equations. Vertex form remains better for reading the turning point and axis of symmetry, so conversion does not make one representation universally superior. It simply exposes different information. Keep the returned a, b, and c together, including zero coefficients. If b is zero, for example, the standard form is still ax² + 0x + c even though the middle term is normally omitted when writing the polynomial by hand.