Problem 37 - College Algebra

Question

Determine the discriminant, and then state how many solutions there are and the nature of the solutions. Do not solve. $$ 6 x^{2}-x-2=0 $$

Step-by-Step Solution

Answer

The discriminant is 49, indicating two distinct real solutions.

Step 1: Identify the quadratic equation format

The given equation is \(6x^2 - x - 2 = 0\). A quadratic equation is typically written in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants.

Step 2: Determine the coefficients

Compare the given equation with the standard format to identify the values of \(a\), \(b\), and \(c\). For this equation, \(a = 6\), \(b = -1\), and \(c = -2\).

Step 3: Calculate the discriminant

The discriminant \(D\) is calculated using the formula \(D = b^2 - 4ac\). Substitute the values of \(a = 6\), \(b = -1\), and \(c = -2\) into the formula: \[D = (-1)^2 - 4(6)(-2)\] Calculate this to find \[D = 1 + 48 = 49\].

Step 4: Analyze the discriminant

With the discriminant \(D = 49\), you check the nature of the solutions: - If \(D > 0\), there are two distinct real solutions. - If \(D = 0\), there is one real solution. - If \(D < 0\), there are two complex solutions. Since \(D = 49\), which is greater than 0, there are two distinct real solutions.

Key Concepts

Discriminant

In quadratic equations, the discriminant is a crucial value that helps us understand the type and number of solutions without actually solving the equation. It is calculated from the coefficients of the quadratic equation in the standard form \(ax^2 + bx + c = 0\). The formula for the discriminant \(D\) is \(D = b^2 - 4ac\).

Here’s why this is important:

• If \(D > 0\), you will have two distinct real solutions.
• If \(D = 0\), there's exactly one real solution, often called a double root.
• If \(D < 0\), the solutions are not real but complex, involving imaginary numbers.

By finding the discriminant, we can immediately know what type of solutions we are dealing with.

Real Solutions

Quadratic equations can have real solutions when the discriminant is greater than or equal to zero. When \(D > 0\), the equation has two real solutions, meaning the parabola representing the equation crosses the x-axis at two different points.

When \(D = 0\), there's one real solution; this means the parabola touches the x-axis at exactly one point, also known as the vertex or "tangent" point. Both cases ensure that the solutions are real numbers that can be plotted on a standard number line.

Complex Solutions

Complex solutions occur in quadratic equations when the discriminant is less than zero. This means \(D < 0\), and the equation does not have real-number solutions because the solutions involve the square root of a negative number.

This results in complex numbers, which are numbers in the form \(a + bi\), where \(i\) is the imaginary unit (\(i^2 = -1\)). Complex solutions always come in conjugate pairs of the form \(a + bi\) and \(a - bi\), which are not plotted as points on a traditional number line but can be represented in the complex plane.

Coefficients

Coefficients in a quadratic equation are the constants \(a\), \(b\), and \(c\) from its standard form \(ax^2 + bx + c = 0\). Each coefficient holds specific roles:

• \(a\), the coefficient of \(x^2\), influences the parabola's width and direction (upward for \(a > 0\) and downward for \(a < 0\)).
• \(b\), tied to the \(x\) term, affects the vertex's horizontal displacement.
• \(c\), the constant term, shifts the parabola vertically and is the y-intercept.

Knowing the values of these coefficients allows us to compute the discriminant and understand the general behavior of the quadratic function on a graph.

Quadratic Formula

The quadratic formula is a powerful tool for finding the solutions (roots) of any quadratic equation. Given an equation in the form \(ax^2 + bx + c = 0\), the solutions for \(x\) can be expressed as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]The plus-minus symbol (\(\pm\)) indicates that there are generally two solutions.

This formula uses the discriminant \(b^2 - 4ac\) to determine the nature of the solutions:

• If \(D > 0\), the roots are real and different.
• If \(D = 0\), the roots are real and equal.
• If \(D < 0\), the roots are complex and conjugate pairs.

By applying the quadratic formula, solutions can be found precisely even when they can't be easily factored.