QRV

Math & Statistics

Quadratic Root Verification Calculator

Evaluate two entered roots in a quadratic, compare them with Vieta's identities, and measure residuals against an entered tolerance.

Coefficient aCoefficient bCoefficient cEntered root 1Entered root 2Residual tolerance

Polynomial value at root 1-

Polynomial value at root 2-

Largest absolute residual-

Expected root sum-

Entered root sum-

Entered minus expected sum-

Expected root product-

Entered root product-

Entered minus expected product-

Largest residual divided by tolerance-

Decision view

Quadratic curve and root residual vectors

Quadratic curve and root residual vectorsBoth candidate roots are plotted on the entered function with direct vertical residuals and Vieta checks.

Exact scenario comparisonEntered root 1 changes while all other entered assumptions remain constant.

Entered root 1	Polynomial value at root 1	Polynomial value at root 2	Largest absolute residual	Expected root sum	Entered root sum	Entered minus expected sum	Expected root product	Entered root product	Entered minus expected product	Largest residual divided by tolerance

How to use Quadratic Root Verification Calculator

Enter coefficients and two candidate roots.
Set a meaningful numerical tolerance.
Read substitution and Vieta diagnostics together.

Calculator guide

Understanding Quadratic Root Verification Calculator

A proposed root should satisfy both direct substitution and the independent sum/product identities.

Substitute root 1 Root 1 has zero default residual.

Substitute root 2 Root 2 also has zero default residual.

Check the root sum The entered pair matches the coefficient-derived sum.

Check the root product The entered pair matches the coefficient-derived product.

Calculation method

How the calculation works

Substitute both entered roots into the polynomial and independently compare their sum and product with Vieta's identities. Substitute each root into ax²+bx+c, then compare their sum with -b/a and their product with c/a.

Detailed calculation process

Verify entered quadratic roots three independent ways

The default verifies roots 2 and 3 for x²-5x+6 with a residual tolerance of 0.0001.

General formula: f(x) = ax^2+bx+ce_1 = f(r_1)e_2 = f(r_2)e_max = max(|e_1|,|e_2|)S_expected = -b/aP_expected = c/aDelta_S = r_1+r_2-S_expectedDelta_P = r_1 r_2-P_expected Substitution checks each root locally. Vieta's identities check the pair using coefficient relationships, providing an independent consistency test.

What each symbol means

a, b, c Quadratic coefficients with a nonzero.

r_1, r_2 Entered candidate roots.

e_1, e_2, e_max Substitution residuals and largest absolute residual.

S_expected, P_expected Expected root sum and product.

Delta_S, Delta_P Entered-minus-expected sum and product differences.

tau Entered residual tolerance.

Worked substitution with the default inputs

1. Substitute root 1 f(2) = 1(2^2)-5(2)+6f(2) = 4-10+6 = 0 Root 1 has zero default residual.

2. Substitute root 2 f(3) = 1(3^2)-5(3)+6f(3) = 9-15+6 = 0 Root 2 also has zero default residual.

3. Check the root sum S_expected = -(-5)/1 = 5r_1+r_2 = 2+3 = 5Delta_S = 0 The entered pair matches the coefficient-derived sum.

4. Check the root product P_expected = 6/1 = 6r_1 r_2 = 2(3) = 6Delta_P = 0 The entered pair matches the coefficient-derived product.

5. Apply the tolerance e_max = max(|0|,|0|) = 0Utilization = 0/0.0001 = 0 All three checks pass the entered tolerance under the defaults.

Both default roots produce zero residual, zero Vieta differences, and zero tolerance utilization.

Purpose-built visual

Quadratic curve with residual vectors

The function curve marks both candidate x-values and draws vertical residual vectors to the x-axis.

Live The chart is regenerated from current inputs.

Units Every axis, marker, and endpoint retains its stated unit.

Check The chart reconciles to the displayed calculation.

Worked situations

Practical examples

The default verifies roots 2 and 3 for x²-5x+6 with a residual tolerance of 0.0001.
Both default roots produce zero residual, zero Vieta differences, and zero tolerance utilization.

Better inputs

Useful tips

Change one input at a time and confirm both the result and visual move.
Keep the units stated beside every field.
Retain intermediate precision and round only the reported result.

Before relying on the result

Limitations and common mistakes

Coefficient a must be nonzero for a quadratic.
A user-entered tolerance determines the numerical pass reference.
Agreement does not prove the intended equation was entered correctly.

Reference

Key terms

Residual: Polynomial value after substituting a candidate root.
Vieta identity: Coefficient relationship for root sum and product.
Tolerance: Entered numerical residual threshold.

Important note

Calculated directly from the entered values using the displayed formula and rounding settings.

Frequently asked questions

Why check Vieta if residuals are zero?

It independently checks the pair.

Can a near-root pass?

That depends on the entered tolerance.

What if a equals zero?

The Vieta quadratic formulas are undefined.

Does zero residual prove data entry?

No.