Problem 49 - Basic Technical Mathematics with Calculus

Question

Find the indicated quadratic equations. Find a quadratic equation for which the solutions are 0.5 and 2.

Step-by-Step Solution

Answer

The quadratic equation is \(x^2 - 2.5x + 1 = 0\).

Step 1: Understand the Relationship

The solutions to a quadratic equation, also known as roots, can be given by the values of \( x \) for which the equation equals zero. These roots can be used to express the factorized form of the quadratic equation.

Step 2: Write Factorized Form Using Roots

Given roots, 0.5 and 2, the factorized form of the quadratic equation can be written as \((x - 0.5)(x - 2) = 0\). These terms in the parentheses are derived directly from the roots by using the formula \(x - \text{root}\).

Step 3: Expand the Factorized Expression

To convert the factorized form \((x - 0.5)(x - 2)\) into the standard quadratic form \(ax^2 + bx + c = 0\), expand the expression:1. Multiply the first terms: \(x \times x = x^2\)2. Multiply the outer terms: \(x \times (-2) = -2x\)3. Multiply the inner terms: \(-0.5 \times x = -0.5x\)4. Multiply the last terms: \(-0.5 \times (-2) = 1\)Combine these to get: \(x^2 - 2x - 0.5x + 1 = 0\).

Step 4: Simplify the Quadratic Equation

Combine like terms in the expression from Step 3:\(x^2 - 2x - 0.5x + 1 = x^2 - 2.5x + 1 = 0\).This is the quadratic equation in the standard form: \(x^2 - 2.5x + 1 = 0\).

Key Concepts

Roots of Quadratic Equation

The roots of a quadratic equation are simply the values of \(x\) that satisfy the equation when set to zero. In simpler terms, if you plug these roots back into the equation, it should balance out to zero. The quadratic equation generally takes the form of \(ax^2 + bx + c = 0\), and the roots represent the points where the parabola of this equation intersects the x-axis. For example, for the given roots 0.5 and 2, if you place them in the equation like \((x - 0.5)(x - 2) = 0\), both values will make the expression zero when solved individually. This is because the roots are obtained by setting each factor to zero: \(x - 0.5 = 0\) yields \(x = 0.5\) and \(x - 2 = 0\) yields \(x = 2\). **Why are roots important?** They help us understand where the graph of the quadratic equation crosses the x-axis.
They're also crucial for converting between different forms of quadratic equations, such as factorized and standard form.

Factorization

Factorization is the process of breaking down an equation into simpler terms or factors that, when multiplied together, give back the original equation. In terms of quadratic equations, factorization involves transforming from the standard form \(ax^2 + bx + c = 0\) to a product of linear factors. Given the roots of the quadratic equation as 0.5 and 2, we express this factorized form as \((x - 0.5)(x - 2) = 0\). **Steps to Factorize:**
1. Identify the roots of the quadratic, in this case, 0.5 and 2.
2. Write the factors as \((x - 0.5)\) and \((x - 2)\).
3. Multiply these factors to find the original quadratic equation. Factorization is particularly useful when you want to solve quadratic equations easily. It's like reverse engineering; you start from the solutions and work your way back to the equation itself.
Understanding factorization allows you to quickly determine the roots just by looking at a factorized form or to solve an equation by setting each factor to zero.

Standard Form of Quadratic Equation

The standard form of a quadratic equation is written as \(ax^2 + bx + c = 0\). This is the most common form and provides a straightforward way to see the equation's coefficients: \(a\), \(b\), and \(c\). The coefficients give us important information about the parabola's orientation and intersections. Starting with the factorized form \((x - 0.5)(x - 2) = 0\), transforming it to the standard form involves expanding the terms back into a single quadratic equation. By multiplying the factors, we first expand the expression:

• Multiply \(x \times x = x^2\)
• Outer terms: \(x \times (-2) = -2x\)
• Inner terms: \(-0.5 \times x = -0.5x\)
• Last terms: \(-0.5 \times (-2) = 1\)

Combining these, the expanded equation becomes \(x^2 - 2.5x + 1 = 0\). This simplified format is useful because it lays out all components of the quadratic equation clearly. It helps in analyzing the graph of the equation and is also necessary for using the quadratic formula. **Remember:** Transforming between forms (factorized to standard) helps you understand each element's contribution to the whole equation.