Problem 91 - 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. $$ 9 x(x-4)=-36 $$

Step-by-Step Solution

Answer

The equation is \(9x^2 - 36x + 36 = 0\), has one real solution, and \(x = 2\).

Step 1: Expand the Given Equation

The first step is to expand the given equation \(9x(x-4) = -36\). Distribute the 9 across the terms in the parenthesis: \[ 9x \cdot x - 9x \cdot 4 = 9x^2 - 36x. \] The equation becomes \[ 9x^2 - 36x = -36. \]

Step 2: Rearrange the Equation

Next, rearrange the equation in the form \(ax^2 + bx + c = 0\) by moving all terms to one side of the equation. Add 36 to both sides: \[ 9x^2 - 36x + 36 = 0. \] Here, \(a = 9\), \(b = -36\), and \(c = 36\).

Step 3: Calculate the Discriminant

Use the formula for the discriminant, \(b^2 - 4ac\), to determine the number of real solutions. Substitute \(a = 9\), \(b = -36\), and \(c = 36\) into the formula: \[ (-36)^2 - 4 \cdot 9 \cdot 36. \] Calculate this value: \[ 1296 - 1296 = 0. \] A discriminant of 0 indicates one real solution.

Step 4: Solve the quadratic using the Quadratic Formula

When the discriminant is 0, we can use the quadratic formula \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] to find the solution. Since \(b = -36\) and the discriminant is 0: \[ x = \frac{36 \pm \sqrt{0}}{18}. \] This simplifies to \[ x = \frac{36}{18} = 2. \] Thus, the equation has one real solution: \(x = 2\).

Key Concepts

Discriminant

The discriminant is a crucial part of the quadratic equation that helps us understand how many real solutions an equation might have. It's calculated using the formula \(b^2 - 4ac\), where \(a\), \(b\), and \(c\) are coefficients from the quadratic equation in the form \(ax^2 + bx + c = 0\).

• If the discriminant is positive, there are two distinct real solutions.
• If it's zero, there is exactly one real solution.
• If it is negative, there are no real solutions since the roots are complex numbers.

For example, when we calculate the discriminant for the equation \(9x^2 - 36x + 36 = 0\), we find it to be 0. This tells us there's just one real solution for the equation.

Quadratic Formula

The quadratic formula is a powerful tool used to find solutions to quadratic equations. Even if factoring by inspection seems impossible, this formula always works. The quadratic formula is given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). It includes:

• \(-b\): Which takes the opposite of the coefficient \(b\).
• \(\pm\sqrt{b^2 - 4ac}\): The plus-minus sign indicates two possible solutions, depending on whether we add or subtract the square root.
• \(2a\): The denominator, which is twice the coefficient of the \(x^2\) term to balance the equation.

When applied, this formula gives solutions for any quadratic equation of the form \(ax^2 + bx + c = 0\). For our solved equation, \(9x^2 - 36x + 36 = 0\), the discriminant was 0, simplifying calculations to one solution: \(x = 2\). This is a result of the term under the square root being zero, thus not affecting \(-b\) when divided by \(2a\).

Real Solutions

In the context of quadratic equations, understanding real solutions is important. These solutions represent the values of \(x\) where the quadratic equation equals zero on a graph.
Real solutions can tell us where the graph of the quadratic function touches or intersects the x-axis. Depending on the discriminant value:

• Two real solutions imply the parabola crosses the x-axis at two points.
• One real solution indicates the parabola touches the x-axis at one point.
• No real solutions mean the parabola does not intersect the x-axis and lies entirely above or below it, indicating complex or imaginary roots.

In our example, since the discriminant was zero, we found there was one real solution, meaning the graph of the equation \(9x^2 - 36x + 36\) touches the x-axis at \(x = 2\). This point is also known as the vertex when the vertex is on the x-axis.