Problem 18 - Algebra 2 and Trigonometry

Question

In \(15-23 :\) a. Find the value of the discriminant and determine if the roots of the quadratic equation are \((1)\) rational and unequal, \((2)\) rational and equal, \((3)\) irrational and unequal, or \((4)\) not real numbers. b. Use any method to find the real roots of the equation if they exist. $$ 2 x^{2}-8=0 $$

Step-by-Step Solution

Answer

The discriminant is 64, so the roots are rational and unequal; they are \(x = 2\) and \(x = -2\).

Step 1: Identify the Coefficients

Given the quadratic equation is \(2x^2 - 8 = 0\). We first need to bring it to standard form \(ax^2 + bx + c = 0\). The given equation can be rewritten as \(2x^2 + 0x - 8 = 0\), where \(a = 2\), \(b = 0\), and \(c = -8\).

Step 2: Calculate the Discriminant

The discriminant \(\Delta\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by \(\Delta = b^2 - 4ac\). Substitute the values of \(a\), \(b\), and \(c\) into the formula: \(\Delta = 0^2 - 4 \cdot 2 \cdot (-8) = 0 + 64 = 64\).

Step 3: Determine the Nature of the Roots

The value of the discriminant is \(64\). Since \(\Delta > 0\) and is a perfect square, the roots of the quadratic equation are real, rational and unequal. Therefore, the roots are of type \((1)\).

Step 4: Solve the Quadratic Equation

We can solve the equation \(2x^2 - 8 = 0\) by factoring or using the quadratic formula. Let's factor: \(2x^2 - 8 = 0\) can be written as \(2(x^2 - 4) = 0\). This further factors to \(2(x - 2)(x + 2) = 0\). Thus, \(x - 2 = 0\) or \(x + 2 = 0\), giving the solutions \(x = 2\) and \(x = -2\).

Step 5: Conclusion on Real Roots

The real roots of the quadratic equation \(2x^2 - 8 = 0\) are \(x = 2\) and \(x = -2\), which are rational and unequal.

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). This type of equation represents a parabola when graphed on the coordinate plane. The value of \( a \) determines the direction and width of the parabola:

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), it opens downwards.
• In our example with the equation \( 2x^2 - 8 = 0 \), it is rewritten to \( 2x^2 + 0x - 8 = 0 \) to fit the standard form, simplifying the problem-solving process.

The solutions to a quadratic equation are the \( x \)-values where the parabola intersects the x-axis, also known as the roots of the equation. These solutions can be found using various methods, such as factoring, completing the square, or using the quadratic formula.

Nature of Roots

The nature of the roots of a quadratic equation is determined by the discriminant, \( \Delta \), which is calculated with the formula \( \Delta = b^2 - 4ac \). The discriminant tells us about the type and number of roots an equation may have:

• If \( \Delta > 0 \), the roots are real and unequal. If it is a perfect square, they are also rational.
• If \( \Delta = 0 \), the roots are real, rational, and equal, meaning the parabola touches the x-axis at one point.
• If \( \Delta < 0 \), the roots are not real numbers, indicating the parabola does not intersect the x-axis.

In the provided equation, \( 2x^2 - 8 = 0 \), we found \( \Delta = 64 \), a positive perfect square, giving us two real, rational, and unequal roots.

Real Roots

Real roots are the solutions of a quadratic equation that can be represented as points on the x-axis of a graph. When an equation's discriminant is positive, it signifies the equation has real roots:

• Real roots can be rational or irrational, depending on whether the discriminant is a perfect square.
• In our example, the equation \( 2x^2 - 8 = 0 \) has a discriminant of 64. This perfect square indicates the roots are real and rational.
• The solutions, \( x = 2 \) and \( x = -2 \), are the points where the parabola intersects the x-axis, demonstrating their status as real roots.

Understanding how to find and classify these roots provides critical insights into the behavior of quadratic equations and their graphs.