Problem 85 - 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. $$ 4 x=x^{2} $$

Step-by-Step Solution

Answer

The solutions are \(x = 4\) and \(x = 0\).

Step 1: Rearrange the Equation

Begin by rearranging the given equation to match the standard form of a quadratic equation \( ax^2 + bx + c = 0 \). Start with the given equation \( 4x = x^2 \). Rewrite it in the form:\[ x^2 - 4x = 0 \]which gives \(a = 1\), \(b = -4\), and \(c = 0\).

Step 2: Calculate the Discriminant

The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula \(b^2 - 4ac\). In this case, substitute \(a = 1\), \(b = -4\), and \(c = 0\) into the formula to get:\[ (-4)^2 - 4 \times 1 \times 0 = 16\]Since the discriminant is 16 (which is greater than zero), there are two distinct real solutions.

Step 3: Solve the Equation

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to solve the equation. With \(a = 1\), \(b = -4\), and \(c = 0\), substitute these into the formula:\[x = \frac{-(-4) \pm \sqrt{16}}{2 \times 1} = \frac{4 \pm 4}{2}\]This gives you:- \(x = \frac{4 + 4}{2} = 4\)- \(x = \frac{4 - 4}{2} = 0\) Thus, the solutions are \(x = 4\) and \(x = 0\).

Key Concepts

Discriminant

In quadratic equations, the discriminant is a key component that indicates the nature of the roots. It's part of the quadratic formula, and it's given by:\[ b^2 - 4ac \]This formula allows you to understand whether the solutions to the quadratic equation are real or complex:

• If the discriminant is greater than zero, there are two distinct real solutions.
• If it equals zero, there is exactly one real solution, also known as a repeated or double root.
• If it's less than zero, the equation has no real solutions; instead, it will have two complex solutions.

Knowing how to calculate and interpret the discriminant helps you quickly determine how many real solutions a quadratic equation has without actually solving it.

Real Solutions

When solving quadratic equations, one of the outcomes we're typically after is the number of "real solutions." Real solutions are simply the values of \(x\) that satisfy the equation and can be plotted on a number line.The number of real solutions is closely linked to the discriminant (\(b^2 - 4ac\)) of the quadratic equation:\[ ax^2 + bx + c = 0 \]

• A positive discriminant suggests two distinct real solutions.
• A zero discriminant implies one real solution (a perfect square). This is because the graph of the equation touches the x-axis at a single point.
• A negative discriminant indicates no real solutions; the graph of the equation does not intersect the x-axis.

For example, in our exercise, the discriminant was calculated as 16, which indicates there are two different real solutions to the equation \(x^2 - 4x = 0\). Hence, understanding real solutions is pivotal in predicting the behavior of quadratic equations.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are any equations that can be written in the form:\[ ax^2 + bx + c = 0 \]The formula itself is:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]It provides a straightforward method to find the values of \(x\) based on the coefficients \(a\), \(b\), and \(c\) from the equation. Here's how it works:

• First, substitute the values of \(a\), \(b\), and \(c\) into the formula.
• Calculate the discriminant \(b^2 - 4ac\) within the formula to determine the nature of the solutions.
• Once the discriminant is known, compute the square root and proceed to find the two potential values for \(x\), using the plus and minus signs \(\pm\) in the formula.

The quadratic formula works universally for any quadratic equation and always leads to the solution, whether the answers are real or complex. In our example, this formula helped solve the rearranged equation \(x^2 - 4x = 0\) by giving us the solutions \(x = 4\) and \(x = 0\). Understanding this formula is essential, as it offers a consistent approach to solving any quadratic equation.