What multiplies to negative 84 and adds to negative 8?

Negative 14 and positive 6, because their product is negative 84 and their sum is negative 8.

What happens when no integer pair exists?

The related quadratic can still have irrational real roots or complex roots.

Product-and-Sum Calculator: Find Factor Pairs & Factor

x · y = P
x + y = S

Factor pairs and quadratic factoring

Find the Number Pair and Factor the Quadratic

Find two numbers that multiply to a product and add to a sum, list every integer pair, explain the signs, solve the related quadratic, or factor ax² + bx + c.

Choose the algebra task

Find a product-and-sum pair Factor or solve a quadratic

Product PSum S

Answer domainInteger search limit

Sign and pair logic

Solution

What multiplies to −84 and adds to −8?

The pair is −14 and 6. Their product is −84 because the signs are different, and their sum is −8 because the negative number has the larger magnitude: −14 + 6 = −8.

Another example: what adds to −9 and multiplies to −252?

The integer pair is −21 and 12: their product is −252 and their sum is −9. Enter any product and sum above instead of relying on a page built for only one pair.

Connection to quadratic equations

If x and y have product P and sum S, then they are the roots of t² − St + P = 0. For P = −84 and S = −8, the equation is t² + 8t − 84 = 0, which factors as (t + 14)(t − 6).

Using the X method calculator for −84 and −8

In the X method, place the product −84 at the top and the sum −8 at the bottom. List opposite-sign factor pairs because the product is negative: −84 and 1, −42 and 2, −28 and 3, −21 and 4, and −14 and 6. Only −14 and 6 add to −8. For the trinomial x² − 8x − 84, write the factors as (x − 14)(x + 6). Expanding checks the result: x² + 6x − 14x − 84 = x² − 8x − 84.

Why signs matter in product-and-sum factoring

A negative product requires one positive number and one negative number. The negative sum means the negative factor must have the greater absolute value. This sign logic reduces the search before any arithmetic is attempted. The Product-and-Sum Calculator lists valid integer pairs and also solves the related quadratic when an integer pair does not exist, so the page supports both X-method factoring and a complete root check.

Product-and-sum questions

What if no integer pair exists?

The calculator still solves the related quadratic and reports real or complex roots according to the selected domain.

Why do opposite signs give a negative product?

A positive number multiplied by a negative number is negative. The sum takes the sign of the number with larger absolute value.

How does this help factor ax² + bx + c?

The ac method searches for numbers whose product is a·c and whose sum is b, then splits the middle term for grouping.

Product-and-Sum Calculator: formula-based step-by-step solution

This Product-and-Sum Calculator worked derivation uses the published default state. Its live formula panel replaces these values whenever an input, unit, mode, date, material, or scenario changes.

Governing formula or algorithm — exact production relationships:

Discriminant D = S^2 - 4P
First number x = (S + √D) ÷ 2
Second number y = (S - √D) ÷ 2
Verification: x + y = S and x × y = P

1. Governing formulas and numeric substitution

Formula 1: Discriminant D = S^2 - 4P
Default-value substitution: Discriminant D = S^2 - 4P | current data: Product P=-84; Sum S=-8; Answer domain=integer; Integer search limit=10000; a=1; b=-8; c=-84; Factoring preference=integer
Formula 2: First number x = (S + √D) ÷ 2
Default-value substitution: First number x = (S + √D) ÷ 2 | current data: Product P=-84; Sum S=-8; Answer domain=integer; Integer search limit=10000; a=1; b=-8; c=-84; Factoring preference=integer
Formula 3: Second number y = (S - √D) ÷ 2
Default-value substitution: Second number y = (S - √D) ÷ 2 | current data: Product P=-84; Sum S=-8; Answer domain=integer; Integer search limit=10000; a=1; b=-8; c=-84; Factoring preference=integer
Formula 4: Verification: x + y = S and x × y = P
Default-value substitution: Verification: x + y = S and x × y = P | current data: Product P=-84; Sum S=-8; Answer domain=integer; Integer search limit=10000; a=1; b=-8; c=-84; Factoring preference=integer

2. Variables and measurement units used

Product P = -84
Sum S = -8
Answer domain = integer
Integer search limit = 10000
a = 1
b = -8
c = -84
Factoring preference = integer

3. Product-and-Sum Calculator final evaluation: this calculator’s production engine evaluates the substituted relationship with unrounded internal precision; its subject-specific result cards, table, and visual report the rounded final answer and verification quantities.

Product-and-sum Calculator: formula, steps, units and verification

product-and-sum calculator is designed for a transparent calculation rather than a black-box answer. Product-and-sum Calculator: find two numbers that multiply to a product and add to a sum. List integer pairs, explain signs, solve roots and factor.

How to use the product-and-sum calculator in 5 steps

Choose the required mode. Select the calculation path that matches the quantity you know and the result you need.

Enter source values. Use measured, documented or assignment values rather than rounded estimates whenever possible.

Confirm every unit. The product-and-sum calculator converts supported units before formula substitution, so each selector must describe the entered number.

Run the calculation. Review the displayed formula, normalized values and numeric substitution before accepting the final result.

Verify the answer. Reproduce the substitution manually and check whether the output is reasonable for the stated assumptions.

Product-and-sum Calculator formula and unit checks

The product-and-sum calculator keeps source inputs, canonical calculation units and displayed output units separate. This prevents a correct formula from producing a wrong answer because feet were treated as meters, percentages as decimals, or time values as the wrong interval.

For an independent check, copy the formula shown by the calculator, substitute the unrounded canonical values, preserve full precision through intermediate operations and round only the final result. The verified answer should match both the displayed number and its measurement unit.

How to interpret the product-and-sum calculator result

A result is useful only when its assumptions match the real problem. Compare the answer with expected ranges, inspect any warning or boundary message, and test a nearby input to confirm that the output changes in the direction predicted by the governing relationship.

The product-and-sum calculator provides an educational and planning result. For regulated, medical, structural, financial, laboratory or safety-critical decisions, verify the inputs and method against the applicable professional requirements before acting.

Solubility Product CalculatorSolubility Product Calculator: for solubility product. Enter the variables, select units, review numeric substitution, reverse solving, validation.

Riemann Sum CalculatorRiemann Sum Calculator: calculate Riemann sum with task-specific inputs, transparent formulas, intermediate steps, unit checks, validation and responsive.

Sum-of-the-Years’-Digits Depreciation CalculatorSum-of-the-years'-digits Depreciation Calculator: for depreciation expense. Enter the variables, select units, review numeric substitution, reverse.

Accent Wall CalculatorAccent Wall Calculator: use U.S. feet and inches to size a board-and-batten, slat or panel accent wall after doors and windows are removed. U.S.-first.

Backsplash Tile CalculatorBacksplash Tile Calculator: for net backsplash area. Enter the variables, select units, review numeric substitution, reverse solving, validation.

Brick CalculatorBrick Calculator: estimate bricks, wall face area, openings, mortar volume, waste and pallets from wall dimensions, brick dimensions and joint thickness.

Product-and-sum Calculator quick verification checklist

Before saving or reporting a result from the product-and-sum calculator, confirm the input source, unit selections, formula mode, intermediate substitution, final unit and rounding rule. These checks make the calculation reproducible and easier to audit.

Browse the complete published calculator sitemap or return to the Salar Cafe home page.

Product-and-Sum Calculator