QFS

Math & Statistics

Quadratic Formula Step-by-Step Calculator

Solve a real-valued quadratic step by step and expose every intermediate term, both root branches, root-sum and root-product checks, plus an entered substitution check.

Coefficient aCoefficient bCoefficient cEntered x for substitutionEntered y comparison

b squared-

Four times a times c-

b squared minus four ac-

Square root of nonnegative discriminant-

Two times a-

Positive-branch root-

Negative-branch root-

Sum of calculated roots-

Product of calculated roots-

Quadratic value at entered x-

Substitution value minus entered y-

Decision view

Quadratic formula node pathway

Quadratic formula node pathwayDiscriminant components split into two root branches before Viete and substitution checks.

Exact scenario comparisonCoefficient c changes while all other entered assumptions remain constant.

Coefficient c	b squared	Four times a times c	b squared minus four ac	Square root of nonnegative discriminant	Two times a	Positive-branch root	Negative-branch root	Sum of calculated roots	Product of calculated roots	Quadratic value at entered x	Substitution value minus entered y

How to use Quadratic Formula Step-by-Step Calculator

Enter coefficients a, b, and c with a nonzero.
Read b squared, 4ac, and the discriminant before the roots.
Follow both numerator branches over the common denominator.
Use Viete and substitution checks to catch sign or arithmetic errors.

Calculator guide

Understanding Quadratic Formula Step-by-Step Calculator

The quadratic formula is easier to audit when b squared, four-ac, the discriminant, square root, denominator, and both branches remain separate. This calculator also checks the roots with Viete's relationships and direct substitution.

Preserve signs Negative b changes both squaring and -b.

One denominator Both branches divide by 2a.

Check twice Viete tests roots against coefficients.

Substitute directly The original polynomial supplies another audit.

Calculation method

How the calculation works

Expose b squared, four-ac, discriminant, square-root term, denominator, both branches, Viete checks, and a direct substitution check as separate audit steps. Calculate b squared and 4ac, subtract for the discriminant, take its nonnegative square root, divide both signed numerator branches by 2a, then verify sum, product, and substitution.

Detailed calculation process

Carry the quadratic formula through both branches and verify it

The defaults solve 2x^2 - 7x + 3 = 0 and also evaluate the polynomial at x = 2.

General formula: Delta=b^2-4ac; x_1=(-b+sqrt(Delta))/(2a); x_2=(-b-sqrt(Delta))/(2a); x_1+x_2=-b/a; x_1x_2=c/a; f(x)=ax^2+bx+c The plus and minus signs create two root branches. Viete's sum and product provide independent coefficient checks, and direct substitution tests any entered x against the original polynomial.

What each symbol means

a, b, c Quadratic coefficients in ax^2+bx+c, with a nonzero.

Delta Discriminant b^2-4ac.

sqrt(Delta) Real square-root term when Delta is nonnegative.

2a Common denominator for both root branches.

x_1, x_2 Positive- and negative-square-root branches.

f(x_c), y_c Polynomial value at comparison x and entered comparison y.

Worked substitution with the default inputs

1. Square b and calculate 4ac: b^2 = (-7)^2 = 49; 4ac = 4*2*3 = 24 Parentheses preserve the negative sign when b is squared.

2. Calculate the discriminant: Delta = 49-24 = 25; sqrt(Delta) = 5 The positive discriminant confirms two real branches.

3. Calculate the denominator: 2a = 2*2 = 4 The same nonzero denominator is used for both roots.

4. Evaluate both branches: x_1 = [7+5]/4 = 3; x_2 = [7-5]/4 = 0.5 Because b is -7, the -b term is positive 7.

5. Run independent checks: Sum = 3+0.5 = 3.5 = -b/a; product = 3*0.5 = 1.5 = c/a; f(2)=8-14+3=-3 The entered comparison y is -3, so the substitution gap is exactly zero.

The default equation has roots 3 and 0.5; their sum and product match -b/a and c/a, and the x = 2 substitution returns the entered y = -3.

Formula pathway

Follow every node from coefficients to checked roots

A node-flow diagram makes the two branches and independent verification paths visible.

Discriminant branch b squared and 4ac meet at subtraction.

Root split Plus and minus numerators separate.

Viete merge Sum and product check both roots together.

Substitution path An entered x returns to the original function.

Worked situations

Practical examples

The default discriminant is 25 and its square root is 5.
The two roots are 3 and 0.5.
Their sum 3.5 and product 1.5 match the coefficient identities.

Better inputs

Useful tips

Keep negative coefficients inside parentheses when squaring.
Carry unrounded intermediates through both branches.
Use complex-number methods when the discriminant is negative.

Before relying on the result

Limitations and common mistakes

The calculator assumes a is nonzero.
It displays real-valued branches only and clips a negative discriminant before the square root in its numeric result fields.
Large or nearly cancelling coefficients may require higher-precision numerical methods.

Reference

Key terms

Branch: One choice of plus or minus in the quadratic formula.
Viete check: Root sum and product identities derived from coefficients.
Substitution gap: Calculated polynomial value minus an entered comparison y.

Important note

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

Frequently asked questions

Why is -b equal to 7 in the default?

The entered b is -7, so negating it produces +7.

What does a negative discriminant mean?

There are no real roots; a complex-number solution is required.

Why check the root product?

For ax^2+bx+c, the exact product must equal c/a.

Why are intermediate values not rounded before the roots?

Keeping full precision through the discriminant, square root, and division avoids introducing avoidable rounding drift.