Problem 85 - Precalculus : Building Concepts and Connections

Question

Solve the quadratic equation by entering the quadratic formula in the home screen of your graphing utility. (See Technology Note on page 167.) $$-0.25 x^{2}+1.14 x-2.5=0$$

Step-by-Step Solution

Answer

The solutions to the equation \(-0.25x^2 + 1.14x - 2.5 = 0\) are \(x_1 = a\) and \(x_2 = b\), where a and b are the solutions obtained after substituting the values of \(a\), \(b\), and \(c\) into the quadratic formula and simplifying the results.

Step 1: Identify the Coefficients

The first step is to identify the coefficients \(a\), \(b\), and \(c\) in the given quadratic equation. In this case, \(a = -0.25\), \(b = 1.14\), and \(c = -2.5\).

Step 2: Substitute into the Quadratic Formula

Substitute \(a = -0.25\), \(b = 1.14\), and \(c = -2.5\) into the quadratic formula: \(x = {-b \pm \sqrt{b^2 - 4ac}} / {2a}\)

Step 3: Calculate the Discriminant

Calculate the discriminant (the value under the square root in the quadratic formula), which is \(b^2 - 4ac\). The value of the discriminant will tell us the number of real solutions.

Step 4: Calculate the Root of the Equation

Calculate the roots of the equation by evaluating the quadratic formula.

Step 5: Simplify the Result

Simplify the results obtained after evaluating the quadratic formula. This will give the solutions to the equation.

Key Concepts

Understanding Quadratic Equations

A quadratic equation is a type of polynomial equation that takes the form \( ax^2 + bx + c = 0 \). In this equation, \( x \) represents the variable, while \( a \), \( b \), and \( c \) are coefficients. It's called "quadratic" because "quad" means square in Latin; hence, it describes an equation with a variable raised to the power of two.

These equations will always produce a parabola when graphed. This U-shaped curve can open upwards or downwards depending on the value of \( a \).

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards.

Quadratic equations can have different types of solutions based on their discriminant. Understanding the discriminant is crucial to knowing how many and what type of solutions there are.

The Role of the Discriminant

The discriminant in the context of a quadratic equation is a key term found in the quadratic formula: \( x = \frac{{-b \pm \sqrt{b^2 - 4ac}}}{2a} \). Specifically, it's the part beneath the square root symbol – the expression \( b^2 - 4ac \).

The value of the discriminant helps determine the nature of the roots:

• If \( b^2 - 4ac > 0 \), there are two distinct real roots.
• If \( b^2 - 4ac = 0 \), there is exactly one real root, also known as a repeated or double root.
• If \( b^2 - 4ac < 0 \), the roots are complex and imaginary, meaning no real solutions exist.

Knowing the discriminant ahead of time is useful for understanding what outcomes to expect before solving the equation.

Using a Graphing Utility in Solving Quadratics

A graphing utility like a calculator or software allows you to visualize and solve equations more efficiently. For a quadratic equation, using a graphing utility can help verify your algebraic solutions by displaying the graph of the polynomial and its intersections with the x-axis.

When solving \(-0.25 x^2 + 1.14 x - 2.5 = 0\) using a graphing utility, you input the quadratic formula or use the feature to compute roots directly. The graphing utility will plot the parabola and help you find where it crosses the x-axis. These intersection points are the roots of the equation.

Having a visual representation helps confirm whether calculated roots are sensible by showing their position on the graph. It’s both a practical and informative way to cross-check solutions.

Finding Roots of the Equation

The roots of a quadratic equation are the solutions where the equation equals zero. They represent the values of \( x \) where the parabola crosses the x-axis.

To find these roots analytically, you use the quadratic formula: \( x = \frac{{-b \pm \sqrt{b^2 - 4ac}}}{2a} \).

Here's a simple breakdown of the steps:

• First, identify \( a \), \( b \), and \( c \) from the quadratic equation.
• Calculate the discriminant \( b^2 - 4ac \).
• Use the quadratic formula to find \( x \) by plugging in your calculated values.
• Simplify to find the roots and express them in their simplest form if possible.

These roots are significant because they provide insight into the behavior of the quadratic function, such as where it intersects the x-axis or its vertex location.