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Quadratic Formula, Discriminant, and Vertex Formulas

Review standard form, the quadratic formula, discriminant, vertex, symmetry axis, intercepts, focus, directrix, and Vieta root checks.

How do the Quadratic Equation Calculator formulas work?

This formula page defines the variables used by the Quadratic Equation Calculator, shows how those values are combined, and explains the order of operations behind the result. Use the worked steps to verify an estimate, compare scenarios, and understand which assumptions or rounding rules can affect the displayed answer.

Formula Overview

The quadratic formula solves every valid equation in standard form. The discriminant classifies the roots, while vertex and conic formulas describe the same function geometrically.

Variable Definitions

  • a is the nonzero coefficient of x².
  • b is the coefficient of x.
  • c is the constant and y-intercept.
  • D is the discriminant b² - 4ac.
  • x₁ and x₂ are the two roots, which may be equal or complex.
  • (h, k) is the vertex.

Standard Form

ax2+bx+c=0,a≠0ax^2+bx+c=0,\quad a\ne0

All terms must be on one side of the equation. When x or the constant term is missing, its coefficient is zero.

Quadratic Formula

x=−b±b2−4ac2ax=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

The plus and minus branches produce two solutions. When the branches match, the equation has one repeated root.

Discriminant and Root Type

D=b2−4acD=b^2-4ac
  • D > 0 produces two distinct real roots.
  • D = 0 produces one repeated real root.
  • D < 0 produces two complex conjugate roots.

Vertex and Axis of Symmetry

h=−b2a,k=f(h),vertex=(h,k)h=-\frac{b}{2a},\quad k=f(h),\quad \text{vertex}=(h,k)
axis of symmetry: x=h\text{axis of symmetry}:\ x=h

If a is positive, the vertex is a minimum. If a is negative, the vertex is a maximum.

Vertex and Factored Forms

y=a(x−h)2+ky=a(x-h)^2+k
y=a(x−x1)(x−x2)y=a(x-x_1)(x-x_2)

Vertex form exposes the turning point. Factored form exposes the roots and remains algebraically meaningful for complex conjugate roots.

Focus, Directrix, and Focal Length

focus=(h,k+14a)\text{focus}=\left(h,k+\frac{1}{4a}\right)
directrix: y=k−14a\text{directrix}:\ y=k-\frac{1}{4a}
focal length=∣14a∣\text{focal length}=\left|\frac{1}{4a}\right|

Vieta Root Checks

x1+x2=−ba,x1x2=cax_1+x_2=-\frac{b}{a},\quad x_1x_2=\frac{c}{a}

These identities provide an independent check of the computed roots without substituting both solutions into the full expression.

Worked Calculation

For 2x² + 5x - 3 = 0, D = 5² - 4(2)(-3) = 49. The roots are (-5 ± 7)/4, so x₁ = -3 and x₂ = 0.50. Their sum is -2.50 = -b/a and their product is -1.50 = c/a.

Formula Assumptions

  • a, b, and c are valid real coefficients and a is nonzero.
  • The equation is written in one variable and standard polynomial form.
  • Complex square roots use i as the imaginary unit.
  • Intermediate calculations are not rounded.
  • Displayed values use the selected two or three decimal places.

Formula FAQ

Why is a not allowed to equal zero?

When a equals zero, the x² term disappears and the equation becomes linear. For a useful Quadratic Equation Calculator formula result, enter realistic values, compare relevant scenarios, check the calculation details, and review the definitions, assumptions, and limitations before using the estimate for an important decision.

Why does a negative discriminant create complex roots?

The square root of a negative real number contains i, producing a conjugate pair. For a useful Quadratic Equation Calculator formula result, enter realistic values, compare relevant scenarios, check the calculation details, and review the definitions, assumptions, and limitations before using the estimate for an important decision.

Can the root formulas be checked another way?

Yes. Substitute each root into the original equation and compare their sum and product with Vieta's formulas. For a useful Quadratic Equation Calculator formula result, use realistic inputs, compare relevant scenarios, and review the stated assumptions before making a decision.

Next Steps

Apply the formulas in the Quadratic Equation Calculator or compare real, repeated, and complex cases in the worked examples.

Use the Quadratic Equation Calculator