Problem 34 - 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. \(x^{2}+6 x=0\)

Step-by-Step Solution

Answer

The discriminant is 36, indicating two distinct real roots: \(x = 0\) and \(x = -6\).

Step 1: Write the Quadratic Equation in Standard Form

First, make sure the given equation is in the standard form of a quadratic equation, which is \(ax^2 + bx + c = 0\). The equation is given as \(x^2 + 6x = 0\). By comparing it with the standard form, we identify that \(a = 1\), \(b = 6\), and \(c = 0\).

Step 2: Calculate the Discriminant

The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is given by \(\Delta = b^2 - 4ac\). Substitute the values of \(a\), \(b\), and \(c\) from the equation. Here, \(b = 6\), \(a = 1\), and \(c = 0\). Therefore, the discriminant is \(\Delta = 6^2 - 4 \times 1 \times 0 = 36\).

Step 3: Determine the Number and Type of Roots

The discriminant helps us determine the nature of the roots of the quadratic equation. If \(\Delta > 0\), there are two distinct real roots. If \(\Delta = 0\), there is one real repeated root. If \(\Delta < 0\), there are two complex roots. Here, \(\Delta = 36 > 0\), indicating that there are two distinct real roots.

Step 4: Apply the Quadratic Formula

The quadratic formula to find the roots of the equation \(ax^2 + bx + c = 0\) is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute \(a = 1\), \(b = 6\), and \(c = 0\) into the formula. Therefore, the roots are:\[x = \frac{-6 \pm \sqrt{36}}{2 \times 1}\]Simplify the expression under the square root:\[x = \frac{-6 \pm 6}{2}\]

Step 5: Calculate the Exact Solutions

Continuing from the solution step above, calculate the two possible values:1. For \(x = \frac{-6 + 6}{2} = 0\).2. For \(x = \frac{-6 - 6}{2} = -6\).Thus, the exact solutions are \(x = 0\) and \(x = -6\).

Key Concepts

Discriminant

The discriminant is a fundamental part of quadratic equations. It is a value that can tell us a lot about the roots, or solutions, of the equation. The discriminant is contained in the standard form of a quadratic equation, which is \(ax^2 + bx + c = 0\).
To calculate it, use the formula \(\Delta = b^2 - 4ac\). The letters \(a\), \(b\), and \(c\) correspond to the coefficients in the quadratic expression.

• If the discriminant \(\Delta > 0\), the quadratic equation has two distinct real roots.
• If \(\Delta = 0\), the equation has exactly one real root, sometimes called a "repeated" or "double" root.
• If \(\Delta < 0\), the quadratic equation has no real roots, but two complex ones.

In the exercise, we found \(\Delta = 36\), indicating two distinct real roots.

Quadratic Formula

The quadratic formula is a powerful tool for finding solutions to quadratic equations. The formula itself is:

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
It uses the coefficients from the standard quadratic form, \(ax^2 + bx + c = 0\), and simplifies the process of finding roots.
Here's how it works:

• \(-b\) is the negation of the linear coefficient.
• \(\pm\) means there are two possible values after the calculation, representing two potential roots.
• \(\sqrt{b^2-4ac}\) represents the discriminant under a square root. This part determines the nature of the roots.
• Divide by \(2a\) which is the doubled value of the quadratic coefficient.

When applied correctly, the quadratic formula gives precise results, as demonstrated in the solution where \(x = 0\) and \(x = -6\) came from plugging \(a = 1\), \(b = 6\), and \(c = 0\) into it.

Real Roots

Real roots are the solutions to a quadratic equation that are real numbers. Whether a quadratic equation has real roots depends on its discriminant value. Real roots can be either distinct or repeated.

• Two distinct real roots occur when the discriminant \(b^2 - 4ac > 0\).
• A repeated real root, also called a double root, happens when \(b^2 - 4ac = 0\).

In this exercise, since the discriminant \(\Delta = 36\), two different real roots were found: \(x=0\) and \(x=-6\). Real roots are important because they represent values where the quadratic graph crosses the x-axis.

Standard Form

The standard form of a quadratic equation is instrumental in solving it, especially when using the quadratic formula or finding the discriminant.
The standard form is written as:

\(ax^2 + bx + c = 0\)
Each coefficient \(a\), \(b\), and \(c\) has its distinct role:

• \(a\) is the coefficient of \(x^2\) and determines the parabola's shape.
• \(b\) is the coefficient of \(x\) and affects the symmetry and position of the parabola.
• \(c\) is the constant term and indicates the point where the parabola intersects the y-axis.

For example, in the given quadratic equation \(x^2 + 6x = 0\), identifying \(a = 1\), \(b = 6\), and \(c = 0\) was essential for using the quadratic formula and calculating the discriminant.