Problem 84 - College Algebra with Modeling and Visualization

Question

Complete the following. (a) Write the equation as \(a x^{2}+b x+c=0\) with \(a>0\) (b) Calculate the discriminant \(b^{2}-4 a c\) and determine the number of real solutions. (c) Solve the equation. $$ 6 x^{2}=4 x $$

Step-by-Step Solution

Answer

The solutions are \(x = 0\) and \(x = \frac{2}{3}\).

Step 1: Convert to Standard Form

Start with the given equation: \(6x^2 = 4x\). To convert it into the standard quadratic form \(ax^2 + bx + c = 0\), subtract \(4x\) from both sides:\[6x^2 - 4x = 0\]Here, \(a = 6\), \(b = -4\), and \(c = 0\). Since \(a = 6 > 0\), this satisfies the requirement that \(a > 0\).

Step 2: Calculate the Discriminant

The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is given by \(b^2 - 4ac\). Substitute \(a = 6\), \(b = -4\), and \(c = 0\) into the formula:\[(-4)^2 - 4 \times 6 \times 0 = 16 - 0 = 16\]Since the discriminant is positive (16 > 0), there are two distinct real solutions.

Step 3: Solve the Quadratic Equation using Factoring

Factor the quadratic equation \(6x^2 - 4x = 0\):First, factor out the greatest common factor, which is \(2x\):\[2x(3x - 2) = 0\]Now, set each factor equal to zero and solve for \(x\):\(2x = 0 \,\Rightarrow\, x = 0\)\(3x - 2 = 0 \,\Rightarrow\, 3x = 2 \,\Rightarrow\, x = \frac{2}{3}\)The solutions are \(x = 0\) and \(x = \frac{2}{3}\).

Key Concepts

Discriminant

The discriminant is a vital part of understanding quadratic equations. It's represented by the formula \(b^2 - 4ac\), where \(a\), \(b\), and \(c\) are the coefficients from the quadratic equation in the form \(ax^2 + bx + c = 0\). The value of the discriminant tells us how many real solutions the quadratic equation has. When calculating the discriminant:

• If the discriminant is greater than zero, there are two distinct real solutions.
• If the discriminant is equal to zero, there is exactly one real solution, also known as a repeated or double root.
• If the discriminant is less than zero, there are no real solutions.

In our exercise, with the equation \(6x^2 - 4x = 0\), we have:

• \(a = 6\)
• \(b = -4\)
• \(c = 0\)

Hence, the discriminant is \((-4)^2 - 4 \times 6 \times 0 = 16\). Since 16 is greater than zero, the equation has two distinct real solutions.

Factoring

Factoring is one of the most common and straightforward ways to solve a quadratic equation. It involves expressing the quadratic equation as a product of its factors, which can then be solved individually.To factor a quadratic equation like \(6x^2 - 4x = 0\), we first look for common factors in each term. In this case:

• We can factor out \(2x\), the greatest common factor of 6x² and -4x.

This gives us:

• \(2x(3x - 2) = 0\)

Next, we apply the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. Thus, we solve:

• \(2x = 0\) leads to \(x = 0\).
• \(3x - 2 = 0\) leads to \(3x = 2\), hence \(x = \frac{2}{3}\).

Through this process, we find the solutions \(x = 0\) and \(x = \frac{2}{3}\). Factoring is effective especially when the quadratic can be easily decomposed into integer factors.

Real Solutions

When solving quadratic equations, identifying real solutions is crucial. Real solutions refer to the values of \(x\) that satisfy the equation and are real numbers, not imaginary ones. The nature of the solutions is primarily determined by the discriminant.In our case, the discriminant is 16, which is greater than zero, indicating two distinct real solutions. These solutions are the x-values where the graph of the quadratic equation \(6x^2 - 4x = 0\) intersects the x-axis.We found these solutions by factoring the equation into \(2x(3x - 2) = 0\) and solving:

• For \(2x = 0\), we get the solution \(x = 0\).
• For \(3x - 2 = 0\), we simplified to get \(x = \frac{2}{3}\).

Therefore, the real solutions to the equation are \(x = 0\) and \(x = \frac{2}{3}\). These solutions show where the quadratic "touches" or "crosses" the x-axis on a graph, providing a visual confirmation of the solutions.