Problem 54 - Precalculus : Building Concepts and Connections

Question

Solve the quadratic equation by using the quadratic formula. Find only real solutions. $$0.25 x^{2}-0.5 x=1$$

Step-by-Step Solution

Answer

The real solutions of the given quadratic equation are \(x = 3\) and \(x = -1\)

Step 1: Identifying the coefficients

First, identify the coefficients a, b and c. In our case \(a = 0.25\), \(b = -0.5\), and \(c = -1\)

Step 2: Calculate the discriminant

The discriminant is defined as \(\triangle = b^{2}-4ac\). This is used to determine the nature of the roots of the quadratic equation. For real solutions, this has to be greater or equal to zero. Calculate the discriminant using the identified coefficients, \(\triangle = (-0.5)^2 - 4*(0.25)*(-1) = 1\)

Step 3: Solve the quadratic equation

Since the discriminant is positive, we can use the quadratic formula to find two real solutions of the equation. We substitute the values of coefficients a, b and c into the formula to get \(x = \frac{-(-0.5) \pm \sqrt{1}}{2*(0.25)}\) which simplifies to \(x = (0.5 \pm 1)\)

Step 4: Calculate the roots of the equation

After simplifying the expressions obtained from the quadratic formula, we find two real roots of the equation: \(x_{1} = (0.5 + 1)/0.5 = 3\) and \(x_{2} = (0.5 - 1)/0.5 = -1\)

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are in the form of ax^2 + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. The formula is given by \[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \]. It provides a method to find the roots (solutions) of any quadratic equation by substituting the coefficients a, b, and c into the formula. The symbol \(\pm\) indicates that there are generally two solutions to a quadratic equation, one with addition and one with subtraction.
When applying the quadratic formula to a problem, you start by identifying the coefficients and substituting them into the formula. Then, the expression under the square root, known as the discriminant, is calculated. This will indicate how many real solutions the equation has, and finally, the roots are computed. The formula is especially useful because it works for all quadratic equations, regardless of whether the roots are real or complex.

Discriminant of a Quadratic Equation

The discriminant of a quadratic equation is a key concept that helps us understand the nature of the roots without actually solving the equation. Denoted as \(\Delta\), it is found within the quadratic formula under the square root, \(\Delta = b^2 - 4ac\). The discriminant can tell us how many solutions a quadratic equation has and whether they are real or complex.

• If \(\Delta > 0\), the equation has two distinct real solutions.
• If \(\Delta = 0\), the equation has exactly one real solution (also called a repeated or double root).
• If \(\Delta < 0\), the equation has two complex solutions.

In the given exercise, the discriminant \(\Delta\) was found to be \(1\), which is greater than zero, so we can conclude that there are two distinct real solutions to the equation. Identifying the value of the discriminant is an essential step before moving forward to actually solving the equation.

Real Solutions of Quadratic Equations

Real solutions of quadratic equations refer to the x-values that satisfy the equation, and when graphed, they are the x-coordinates where the parabola intersects the x-axis. A quadratic equation may have up to two real solutions, one real solution, or none at all, depending on the discriminant. As we previously noted, a discriminant greater than zero results in two real solutions, equal to zero results in one real solution, and less than zero means there are no real solutions, only complex ones.
In our sample exercise, the discriminant was calculated as \(1\), indicating two real solutions. These solutions are found by solving the quadratic equation using the quadratic formula. The final roots for our equation were \(x_1 = 3\) and \(x_2 = -1\), which means that the quadratic equation given by \(0.25 x^2 - 0.5 x = 1\) has two points where it intersects the x-axis.

Solving Quadratic Equations Step by Step

Solving quadratic equations step by step involves a methodical approach to find the roots with precision. Here's how you can tackle a quadratic equation problem:

• Step 1: Write down the equation in standard form \(ax^2 + bx + c = 0\) and identify the coefficients a, b, and c.
• Step 2: Calculate the discriminant \(\Delta = b^2 - 4ac\) to determine the nature of the roots.
• Step 3: Use the quadratic formula with the coefficients to find the roots of the equation. Depending on the value of \(\Delta\), expect either two real solutions, one real solution, or two complex solutions.
• Step 4: Simplify the roots obtained from the quadratic formula to find the most simplified form of the real solutions.

Following these steps ensures a systematic approach to solving quadratic equations. In our example, after calculating the discriminant (Step 2), we used the quadratic formula (Step 3) and simplified the results to find the roots \(x_1 = 3\) and \(x_2 = -1\) (Step 4), thereby solving the equation completely.