Solve Each Of The Quadratic Equations 3x 0.5 X2

Solving the Quadratic Equation: 3x - 0.5x² = 0

This article provides a comprehensive guide to solving the quadratic equation 3x - 0.5x² = 0. We'll explore various methods, delve into the underlying concepts, and offer practical applications to solidify your understanding. Understanding quadratic equations is fundamental in many areas, from physics and engineering to finance and computer science.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (typically 'x') is 2. The general form is ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero (otherwise, it wouldn't be a quadratic equation). Our specific equation, 3x - 0.5x² = 0, fits this form, though it might appear slightly different at first glance.

Identifying a, b, and c in our Equation

To solve our equation effectively, let's rewrite it in the standard form:

-0.5x² + 3x + 0 = 0

Now we can clearly identify the coefficients:

• a = -0.5
• b = 3
• c = 0

The fact that 'c' is zero simplifies the solution process considerably, as we'll see.

Methods for Solving Quadratic Equations

Several methods exist for solving quadratic equations. The most common are:

• Factoring: This involves expressing the quadratic equation as a product of two linear expressions.
• Quadratic Formula: A direct formula that provides the solutions for any quadratic equation.
• Completing the Square: A method that manipulates the equation to create a perfect square trinomial.
• Graphing: Visualizing the equation's parabola to find its x-intercepts (roots).

We'll explore the most efficient methods for our specific equation, focusing on factoring and the quadratic formula.

Solving 3x - 0.5x² = 0 by Factoring

Factoring is often the quickest method when it's applicable. Since our equation has a 'c' value of 0, factoring becomes particularly straightforward. Notice that both terms on the left-hand side of the equation contain 'x':

3x - 0.5x² = 0

We can factor out 'x':

x(3 - 0.5x) = 0

This equation is now in a factored form. The product of two factors is zero if and only if at least one of the factors is zero. Therefore, we set each factor equal to zero and solve for x:

Factor 1: x = 0

Factor 2: 3 - 0.5x = 0 => 0.5x = 3 => x = 6

Therefore, the solutions to the equation 3x - 0.5x² = 0 are x = 0 and x = 6.

Solving 3x - 0.5x² = 0 using the Quadratic Formula

The quadratic formula is a universal method applicable to all quadratic equations, regardless of whether they're easily factorable. The formula is:

x = (-b ± √(b² - 4ac)) / 2a

Substituting our values (a = -0.5, b = 3, c = 0) into the quadratic formula:

x = (-3 ± √(3² - 4 * (-0.5) * 0)) / (2 * -0.5)

x = (-3 ± √9) / -1

x = (-3 ± 3) / -1

This leads to two solutions:

Solution 1: x = (-3 + 3) / -1 = 0 / -1 = 0

Solution 2: x = (-3 - 3) / -1 = -6 / -1 = 6

Again, we arrive at the solutions x = 0 and x = 6.

Graphical Representation

The graph of the quadratic equation y = 3x - 0.5x² is a parabola opening downwards. The x-intercepts of this parabola represent the solutions to the equation 3x - 0.5x² = 0. These intercepts occur at x = 0 and x = 6, visually confirming our algebraic solutions. (Note: Creating the graph is recommended but falls outside the scope of this text-based response).

Interpreting the Solutions

The solutions x = 0 and x = 6 represent the points where the parabola intersects the x-axis. These points are also known as the roots or zeros of the quadratic equation. In practical applications, these roots could represent, for instance:

• Break-even points in business: If the equation models profit (y) as a function of production (x), the roots represent the production levels where profit is zero (break-even).
• Time points in physics: If the equation describes the trajectory of a projectile, the roots might signify the time at which the projectile is at ground level.
• Equilibrium points in systems: In many systems, the solutions to a quadratic equation can describe points of equilibrium or stability.

The specific interpretation depends heavily on the context in which the quadratic equation is used.

Further Exploration: Discriminant and Nature of Roots

The expression inside the square root in the quadratic formula (b² - 4ac) is called the discriminant. It provides valuable information about the nature of the roots:

• Discriminant > 0: Two distinct real roots (as in our example).
• Discriminant = 0: One real root (a repeated root).
• Discriminant < 0: Two complex roots (involving imaginary numbers).

In our case, the discriminant is 9 (3² - 4 * (-0.5) * 0 = 9), which is greater than zero, indicating two distinct real roots.

Advanced Applications and Extensions

The principles demonstrated here extend to more complex quadratic equations and related mathematical concepts. Understanding quadratic equations is crucial for tackling:

• Solving higher-order polynomial equations: Techniques like factoring and the quadratic formula form the basis for solving cubic and quartic equations.
• Solving systems of equations: Quadratic equations often appear within systems of equations that require simultaneous solutions.
• Calculus: Derivatives and integrals of quadratic functions play a vital role in calculus.
• Optimization problems: Quadratic equations are often used to find the maximum or minimum value of a function.

This comprehensive guide equipped you with the knowledge and techniques to confidently solve quadratic equations like 3x - 0.5x² = 0. Remember to practice these methods to solidify your understanding and prepare for more advanced applications. Through consistent practice and exploration, you'll gain mastery over this essential mathematical concept.