Problem 38 - Algebra 2

Question

Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula. \(.4 x^{2}+x-0.3=0\)

Step-by-Step Solution

Answer

Discriminant is 1.48; two distinct real roots; solutions are approximately 0.27 and -2.70.

Step 1: Identify Coefficients

For the quadratic equation \(0.4x^2 + x - 0.3 = 0\), identify the coefficients: \(a = 0.4\), \(b = 1\), and \(c = -0.3\).

Step 2: Calculate Discriminant

The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula \(b^2 - 4ac\). Substitute the values to find the discriminant: \(1^2 - 4 \times 0.4 \times (-0.3) = 1 + 0.48 = 1.48\).

Step 3: Interpret Discriminant

Since the discriminant \(1.48\) is greater than zero, the quadratic equation has two distinct real roots.

Step 4: Apply Quadratic Formula

The solutions of the quadratic equation \(ax^2 + bx + c = 0\) can be found using the Quadratic Formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute the values: \(x = \frac{-1 \pm \sqrt{1.48}}{2 \times 0.4}\).

Step 5: Simplify Solutions

Calculate \(\sqrt{1.48} \approx 1.216\). The two solutions are: \(x = \frac{-1 + 1.216}{0.8}\) and \(x = \frac{-1 - 1.216}{0.8}\). This gives us \(x \approx 0.27\) and \(x \approx -2.70\).

Key Concepts

Discriminant

In a quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant plays a crucial role in determining the nature of the roots. The discriminant is given by the formula \(b^2 - 4ac\). This value can tell us whether the roots are real or complex, and whether they are distinct or repeated.

In our example, using the equation \(0.4x^2 + x - 0.3 = 0\), we identify the coefficients \(a = 0.4\), \(b = 1\), and \(c = -0.3\). When we calculate the discriminant, we find it to be \(1.48\).

The value of the discriminant can help us as follows:

• If the discriminant is greater than zero, like our \(1.48\), there are two distinct real roots.
• If it is equal to zero, the quadratic has one repeated real root.
• If it is less than zero, the roots are complex and come in a conjugate pair.

This calculation is fundamental in determining the subsequent steps for solving the quadratic equation.

Quadratic Formula

Once the discriminant has been calculated, the next step is to find the roots using the quadratic formula. This is perhaps the most widely used method to find solutions to quadratic equations. The quadratic formula is expressed as:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Let's see how it applies to our equation \(0.4x^2 + x - 0.3 = 0\). We substitute \(a = 0.4\), \(b = 1\), and \(c = -0.3\) into the quadratic formula as follows:

• Calculate the square root of the discriminant \(\sqrt{1.48} \approx 1.216\).
• Substitute into the formula: \(x = \frac{-1 \pm 1.216}{0.8}\).

The "\(\pm\)" in the formula symbolizes that there are two possible solutions, one for the plus and one for the minus. So this formula not only finds the roots but narrows it down to the exact values.

This formula provides a reliable method to determine the roots of any quadratic equation as long as you identify your \(a\), \(b\), and \(c\) correctly.

Real Roots

Real roots of a quadratic equation are the solutions where the equation equals zero and are actual numbers you can plot on a real number line. Recognizing and interpreting real roots is essential.

In our scenario, since the discriminant \(1.48\) is positive, it ensures that the quadratic equation \(0.4x^2 + x - 0.3 = 0\) will possess two distinct real roots.

Using the quadratic formula, we found the approximate values of these roots to be \(x \approx 0.27\) and \(x \approx -2.70\). These values mean that the graph of this quadratic equation would intersect the x-axis at these points.

When the roots are real:

• The graph of the equation is a parabola that either opens upwards or downwards.
• The points where the parabola intersects the x-axis are the roots.

Understanding real roots helps in graphically analyzing the behavior of quadratic functions and predicting their intersections with the axes.