Quadratic formula Equation, Derivation & Rearrangements

Core idea

Overview

The quadratic formula provides the roots of a second-degree polynomial equation in the form ax² + bx + c = 0. It is derived from the algebraic process of completing the square and describes the specific values of the independent variable where the parabola intersects the horizontal axis.

When to use: Use this formula when dealing with a quadratic equation that cannot be easily factored by inspection. It is applicable whenever the equation is expressed in standard form and the leading coefficient 'a' is not zero.

Why it matters: This formula is fundamental in physics for calculating projectile motion and in engineering for optimizing structural designs. It allows for the identification of real or complex solutions, providing essential insight into the behavior of parabolic systems.

Symbols

Variables

a = Coefficient a, b = Coefficient b, c = Coefficient c, x = Root (one solution)

a

Coefficient a

Variable

b

Coefficient b

Variable

c

Coefficient c

Variable

x

Root (one solution)

Variable

Walkthrough

Derivation

Derivation of the Quadratic Formula

The quadratic formula gives the exact solutions to any general quadratic equation. It is derived by completing the square on the standard form.

The equation is in the form ax² + bx + c = 0.
The coefficient a is not equal to zero.

1

Start with Standard Form:

Begin with the general form of a quadratic equation.

a x^{2} + b x + c = 0

2

Divide by a:

Divide the entire equation by a so the leading coefficient becomes 1, making it easier to complete the square.

x^{2} + \frac{b}{a} x + \frac{c}{a} = 0

3

Complete the Square:

Halve the x-coefficient and square the bracket, subtracting the square of the added term to keep the equation balanced.

(x + \frac{b}{2 a})^{2} - (\frac{b}{2 a})^{2} + \frac{c}{a} = 0

4

Rearrange the Constants:

Move constant terms to the right and combine them using a common denominator.

(x + \frac{b}{2 a})^{2} = \frac{b ^{2}}{4 a ^{2}} - \frac{c}{a} = \frac{b ^{2} - 4 a c}{4 a ^{2}}

5

Take the Square Root:

Square-root both sides, remembering to include both positive and negative roots.

x + \frac{b}{2 a} = \pm \frac{b ^{2} - 4 a c}{4 a ^{2}} = \pm \frac{b ^{2} - 4 a c}{2 a}

6

Isolate x:

Subtract $\frac{b}{2 a}$ from both sides to obtain the quadratic formula.

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

Result

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

Source: Standard curriculum — A-Level Pure Mathematics

Visual intuition

Graph

The graph is a hyperbola because the variable appears in the denominator of the formula. It features a vertical asymptote at the origin and curves that approach the x-axis as the input value becomes very large or very small.

Graph type: hyperbolic

Why it behaves this way

Intuition

The quadratic formula geometrically describes finding the x-coordinates where a parabolic curve, represented by y = ax2 + bx + c, intersects the horizontal x-axis.

a

Coefficient of the quadratic term (x2), determining the parabola's concavity and vertical stretch.

If 'a' is positive, the parabola opens upwards; if negative, it opens downwards. A larger absolute value of 'a' makes the parabola narrower.

b

Coefficient of the linear term (x), influencing the horizontal position of the parabola's vertex.

Changes in 'b' shift the parabola horizontally and vertically, affecting the location of its axis of symmetry.

c

Constant term, representing the y-intercept of the parabola.

This is the point where the parabola crosses the vertical axis (when x=0).

x

The roots or solutions of the quadratic equation.

These are the specific x-values where the parabola intersects the horizontal axis (where y=0).

Signs and relationships

±: The plus-minus sign indicates that a quadratic equation generally has two distinct solutions (roots), corresponding to the two points where a parabola can intersect the x-axis.
b2-4ac: This term, known as the discriminant, determines the nature of the roots. If positive, there are two distinct real roots; if zero, one real root (a repeated root); if negative, two complex conjugate roots.
√: The square root operation arises from the algebraic process of completing the square and ensures that the solutions account for both positive and negative possibilities when solving for x2-like terms.
-b: The negative sign here, along with the '2a' in the denominator, positions the axis of symmetry of the parabola at x = -b/(2a).

Free study cues

Insight

Canonical usage

The quadratic formula requires all terms in the standard form equation (ax2 + bx + c = 0) to have the same physical dimension for dimensional consistency.

Common confusion

A common mistake is to overlook the dimensional consistency of the coefficients 'a', 'b', and 'c' with respect to the variable 'x', particularly in applied problems where 'x' represents a physical quantity.

Unit systems

$x$ Varies by context (e.g., meters, seconds, dimensionless) · The fundamental dimension of the variable being solved for.

$c$ Varies by context · The dimension of the constant term, which dictates the common dimension for all terms in the equation.

$b$ Varies by context · The dimension of the coefficient of the linear term, such that the product [b][x] has the dimension [D].

$a$ Varies by context · The dimension of the coefficient of the quadratic term, such that the product [a][x]2 has the dimension [D].

One free problem

Practice Problem

Find the larger root of the equation x² - 7x + 10 = 0.

Coefficient a1

Coefficient b-7

Coefficient c10

Solve for: $x 1$

Hint: The discriminant b² - 4ac is equal to 9, which is a perfect square.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

When finding the time a projectile hits the ground, Quadratic formula is used to calculate Root from Coefficient a, Coefficient b, and Coefficient c. The result matters because it helps connect the calculation to the shape, rate, probability, or constraint in the model.

Study smarter

Tips

Calculate the discriminant (b² - 4ac) first to determine the nature of the roots.
Ensure the equation equals zero before identifying the coefficients a, b, and c.
Carefully maintain the signs of the coefficients during substitution into the formula.
Remember that the ± symbol indicates the potential for two distinct solutions.

Avoid these traps

Common Mistakes

Wrong sign for -b.
Computing b²-4ac incorrectly.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

The quadratic formula gives the exact solutions to any general quadratic equation. It is derived by completing the square on the standard form.

Use this formula when dealing with a quadratic equation that cannot be easily factored by inspection. It is applicable whenever the equation is expressed in standard form and the leading coefficient 'a' is not zero.

This formula is fundamental in physics for calculating projectile motion and in engineering for optimizing structural designs. It allows for the identification of real or complex solutions, providing essential insight into the behavior of parabolic systems.

Wrong sign for -b. Computing b²-4ac incorrectly.

When finding the time a projectile hits the ground, Quadratic formula is used to calculate Root from Coefficient a, Coefficient b, and Coefficient c. The result matters because it helps connect the calculation to the shape, rate, probability, or constraint in the model.

Calculate the discriminant (b² - 4ac) first to determine the nature of the roots. Ensure the equation equals zero before identifying the coefficients a, b, and c. Carefully maintain the signs of the coefficients during substitution into the formula. Remember that the ± symbol indicates the potential for two distinct solutions.

References

Sources

Wikipedia: Quadratic formula
Stewart, J. (2015). Calculus: Early Transcendentals. Cengage Learning.
Halliday, Resnick, Walker - Fundamentals of Physics, 10th Edition
Britannica: Quadratic equation
Wikipedia: Quadratic equation
Standard curriculum — A-Level Pure Mathematics

Quadratic formula

Overview

Variables

Derivation

Start with Standard Form:

Divide by a:

Complete the Square:

Rearrange the Constants:

Take the Square Root:

Isolate x:

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Discriminant

Frequently Asked Questions

Sources