Quadratic Formula Calculator and Guide

Welcome to the Numplexis quadratic formula calculator and guide! The following contents contain a quadratic formula calculator, it's complementary excel spreadsheet, and a step by step derivation of the quadratic formula from the quadratic equation.

Quadratic Formula Calculator

To use the calculator, edit the input boxes below given your quadratic constants a, b, and c.

Quadratic Equation Standard Form:

ax2 + bx + c = 0
a
b
c

Quadratic Equation Input:

Quadratic Calculator Output:

Download Quadratic Excel Spreadsheet

Calculator Background Information

Used in projectile motion and a host of numerical applications, the quadratic formula is a simple yet formidable tool for solving systems of equations. The generalized application begins with a set of equations with two independent variables. The typical end goal involves determining an equilibrium state or condition in which the equalities of each equation in the system are satisfied by specific quantities substituted for each independent variable. Determining which quantities to substitute is where the quadratic formula comes into play. Most often the solution involves equating one independent variable in terms of the other, substituting the relationship into the remaining equation to achieve one equation with one independent variable, and then simplify it to the quadratic form. From there it is a matter of employing the quadratic formula to achieve the desired solution.

Deriving the Quadratic Formula

Using the quadratic formula is analogous to taking a shortcut. The path we bypass entails a series of algebraic operations followed to reach the end result we seek. The reason such a shortcut exists is the consistency and repeatability for which the the algebra affords us in the derivation. To better appreciate what the quadratic formula buys us, we walk through the derivation below.


Equation 1. Standard Form Quadratic Equation
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To attain equation 2: Divide each side of the equation by the constant a.


Equation 2. Standard Formula Derivation, Step 1
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To attain equation 3: Subtract the c/a term to each side of the equation.


Equation 3. Standard Formula Derivation, Step 2
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To attain equation 4: Add the (b/2a)2 term to each side of the equation.


Equation 4. Standard Formula Derivation, Step 3
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To attain equation 5: Factor the left side of the equation to encompass one square term (completing the square). On the right side of the equation, distribute the 2nd power to the numerator and denominator of the b/2a term.


Equation 5. Standard Formula Derivation, Step 4
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To attain equation 6: Multiply numerator and denominator of -c/a by 4a to achieve a common denominator on the right side of the equation.


Equation 6. Standard Formula Derivation, Step 5
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To attain equation 7: Take the square root of each side of the equation.


Equation 7. Standard Formula Derivation, Step 6
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To attain equation 8: Extract the denominator from the square root on the right side of the equation.


Equation 8. Standard Formula Derivation, Step 7
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To solve for equation 9: Solve for x by adding b/2a to each side of the equation.


Equation 9. Quadratic Formula
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