Problem 89 - 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. $$ 2 x^{2}+3 x=12-2 x $$

Step-by-Step Solution

Answer

The equation is \(2x^2 + 5x - 12 = 0\) with solutions \(x = \frac{3}{2}\) and \(x = -4\).

Step 1: Rewrite the equation

Start by moving all terms to one side of the equation to have zero on the other side. The given equation is:\[ 2x^2 + 3x = 12 - 2x \]Add \(2x\) to both sides and subtract 12 from both sides to get:\[ 2x^2 + 5x - 12 = 0 \]Thus, the equation is now in the form \( ax^2 + bx + c = 0 \) where \( a = 2 \), \( b = 5 \), and \( c = -12 \).

Step 2: Calculate the Discriminant

The discriminant for a quadratic equation \( ax^2 + bx + c = 0 \) is calculated using the formula \( b^2 - 4ac \).Substitute \( a = 2 \), \( b = 5 \), and \( c = -12 \) into the formula:\[ b^2 - 4ac = 5^2 - 4(2)(-12) \]\[ = 25 + 96 \]\[ = 121 \]The discriminant is 121.

Step 3: Determine the Number of Real Solutions

The sign of the discriminant helps determine the number of real solutions: - If the discriminant is positive, there are two distinct real solutions. - If it is zero, there is exactly one real solution. - If it is negative, there are no real solutions. Since the discriminant is 121, which is positive, there are two distinct real solutions.

Step 4: Solve the Equation Using Quadratic Formula

To find the solutions of the equation \( 2x^2 + 5x - 12 = 0 \), use the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Plug in \( a = 2 \), \( b = 5 \), and \( c = -12 \):\[ x = \frac{-5 \pm \sqrt{121}}{4} \]Since \( \sqrt{121} = 11 \), we have:\[ x = \frac{-5 + 11}{4} \] or \[ x = \frac{-5 - 11}{4} \]This gives \( x = \frac{6}{4} = \frac{3}{2} \) and \( x = \frac{-16}{4} = -4 \).Therefore, the solutions are \( x = \frac{3}{2} \) and \( x = -4 \).

Key Concepts

Discriminant

When it comes to quadratic equations, one of the most useful tools to determine the nature of the solutions is the discriminant. The formula for the discriminant is represented as \( b^2 - 4ac \). This simple calculation gives valuable insights into the nature of the solutions you can expect from the equation.

• If the discriminant is a positive number, it means the quadratic equation has two distinct real solutions.
• If the discriminant equals zero, the equation has exactly one real solution, which is also called a repeated or double root.
• If the discriminant is negative, it implies that there are no real solutions; instead, you would have two complex or imaginary solutions.

In our example, we calculated the discriminant as 121, which is clearly positive. Therefore, we know we will have two real solutions.

Real Solutions

Real solutions are solutions to equations that are not imaginary or complex. They can be real numbers that you could, for instance, plot on a number line. When solving a quadratic equation, determining the type of solutions depends directly on the discriminant, as explained above.
A positive discriminant indicates the presence of two distinct real solutions. This means that the parabola represented by the quadratic equation will intersect the x-axis at two points.

The example from our exercise yields a positive discriminant. Thus, the two real solutions, which are \( x = \frac{3}{2} \) and \( x = -4 \), intersect the x-axis at these points. These real values are practical and help you understand the behavior of the quadratic function in real-world scenarios.

Quadratic Formula

To solve a quadratic equation, especially when factoring is not straightforward, the quadratic formula becomes extremely valuable. The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula allows you to calculate the solutions of any quadratic equation by substituting the values of \( a \), \( b \), and \( c \). Here is a quick guide on how you use it:

• Ensure your equation is arranged in the format \( ax^2 + bx + c = 0 \).
• Plug the coefficients \( a \), \( b \), and \( c \) into the formula.
• Calculate the discriminant \( b^2 - 4ac \) found in the square root.
• Solve for \( x \) using both the "+" and "-" operations to find the two potential solutions.

Using our example equation \( 2x^2 + 5x - 12 = 0 \), we've already calculated values and found solutions of \( x = \frac{3}{2} \) and \( x = -4 \), illustrating how effective and robust the quadratic formula is for solving quadratics.