Calculate The Following A The Value of The Linear Quadratic

A linear quadratic equation combines linear and quadratic terms to model relationships between variables. This guide explains how to calculate its value, including the formula, assumptions, and practical applications.

What is a Linear Quadratic Equation?

A linear quadratic equation is a mathematical expression that contains both linear (first-degree) and quadratic (second-degree) terms. It typically appears in the form:

ax² + bx + c = 0

Where:

a is the coefficient of the quadratic term (x²)
b is the coefficient of the linear term (x)
c is the constant term

These equations are fundamental in algebra and have applications in physics, engineering, and economics.

How to Calculate the Value

To find the value of a linear quadratic equation, you can use the quadratic formula:

x = [-b ± √(b² - 4ac)] / (2a)

This formula provides the roots of the equation, which are the values of x that satisfy the equation.

Steps to Calculate

Identify the coefficients a, b, and c in the equation
Calculate the discriminant (b² - 4ac)
If the discriminant is positive, there are two real roots
If the discriminant is zero, there is one real root
If the discriminant is negative, there are no real roots (complex roots exist)
Apply the quadratic formula to find the roots

The Formula

The quadratic formula is derived from completing the square and solving for x:

ax² + bx + c = 0 x² + (b/a)x + (c/a) = 0 x² + (b/a)x = -(c/a) x² + (b/a)x + (b/2a)² = -(c/a) + (b/2a)² (x + b/2a)² = (b² - 4ac)/4a² x + b/2a = ±√(b² - 4ac)/2a x = [-b ± √(b² - 4ac)] / (2a)

This formula is valid for any quadratic equation where a ≠ 0.

Worked Example

Let's solve the equation 2x² + 4x - 6 = 0:

Identify coefficients: a = 2, b = 4, c = -6
Calculate discriminant: (4)² - 4(2)(-6) = 16 + 48 = 64
Since discriminant is positive, there are two real roots
Apply quadratic formula:
x = [-4 ± √64] / (2*2) x = [-4 ± 8] / 4
Calculate both roots:
x₁ = (-4 + 8)/4 = 4/4 = 1 x₂ = (-4 - 8)/4 = -12/4 = -3

The solutions are x = 1 and x = -3.

FAQ

What is the difference between linear and quadratic equations?

Linear equations have terms with variables raised to the first power (e.g., 2x + 3 = 0), while quadratic equations have terms with variables raised to the second power (e.g., x² + 2x + 1 = 0).

When would I use a linear quadratic equation?

You would use a linear quadratic equation when modeling situations where both linear and quadratic relationships exist, such as projectile motion, optimization problems, or growth/decay models.

What does the discriminant tell me about the roots?

The discriminant (b² - 4ac) indicates the nature of the roots: positive means two real roots, zero means one real root, and negative means two complex roots.