Problem 25 - A Graphical Approach to College Algebra

Question

Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}=2 x+24$$

Step-by-Step Solution

Answer

The real solutions are \(x = 6\) and \(x = -4\).

Step 1: Move all terms to one side

Start by rewriting the given equation: \[ x^2 = 2x + 24 \]Subtract \(2x\) and \(24\) from both sides to set the equation to zero:\[ x^2 - 2x - 24 = 0 \]

Step 2: Factor the quadratic equation

Look for two numbers that multiply to \(-24\) and add to \(-2\). These numbers are \(-6\) and \(4\). Write the equation as:\[ (x - 6)(x + 4) = 0 \]

Step 3: Solve for x

Set each factor equal to zero and solve for \(x\):\[ x - 6 = 0 \Rightarrow x = 6 \]\[ x + 4 = 0 \Rightarrow x = -4 \]The solutions are \(x = 6\) and \(x = -4\).

Step 4: Verify solutions graphically

Graph the equation \(y = x^2 - 2x - 24\). The solutions are the x-intercepts of the graph, which occur where \(y = 0\). Verify that the x-intercepts are at \(x = 6\) and \(x = -4\).

Key Concepts

Factoring Quadratics

Factoring quadratics is a powerful technique to solve quadratic equations in the form of \(ax^2+bx+c=0\). The goal is to express the quadratic equation as a product of linear factors.For instance, in the original exercise, we had the equation \(x^2 - 2x - 24 = 0\). To factor it, you find two numbers that multiply to \(-24\) (the constant term) and add to \(-2\) (the coefficient of the linear term). After thinking through possible pairs, you find that \(-6\) and \(4\) work. This way, the quadratic expression becomes \((x - 6)(x + 4) = 0\). The beauty of factoring is:

• It converts the quadratic equation into simpler linear equations.
• It gives you the solutions directly when each factor is set to zero.

In many cases, this is a quick and efficient method. However, it works best when the factors are easy to spot. Otherwise, other methods like completing the square or using the quadratic formula may be necessary.

Graphical Solution

In solving quadratic equations, the graphical solution offers a visual understanding of the equation's roots. By graphing the equation \(y = x^2 - 2x - 24\), you look for points where the graph intersects the x-axis. These intersection points are the solutions to the original equation, also known as the x-intercepts. Graphically, this process helps:

• Confirm the solutions calculated algebraically.
• Understand the nature of solutions. If you see two x-intercepts, there are real solutions. No x-intercepts mean no real solutions.
• Visualize how changes in the equation affect the graph and the roots.

Using graphing technology or plotting the graph manually are practical ways to employ this method, ensuring your algebraic solution is correct.

Quadratic Formula

The quadratic formula is a universal tool to find solutions of any quadratic equation, regardless of whether it can be factored easily or not.Given any quadratic equation in the form \(ax^2 + bx + c = 0\), the solutions can be found using the formula:\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \]This formula always works and provides solutions even when:

• The equation cannot be factored directly or easily.
• There are complex or irrational roots involved.

In our exercise, using the quadratic formula on \(x^2 - 2x - 24\) confirms the same solutions achieved by factoring, but it's especially useful when factoring isn't straightforward. It's a helpful method to ensure accuracy across various equation types.

X-intercepts

X-intercepts are crucial in understanding the behavior of quadratic graphs in relation to their equations. They represent the values of \(x\) where the graph hits or crosses the x-axis, meaning \(y=0\) at these points. In our solved equation, the solutions were \(x=6\) and \(x=-4\), which are precisely the x-intercepts of the graph of \(y=x^2-2x-24\).These intercepts are essential because:

• They reveal the real solutions to the quadratic equation.
• They show where a projectile, in applications, would land on a horizontal surface.
• They help in determining the range and roots visually on a graph.

By understanding and finding x-intercepts, you gain insight not only into the solutions but also into the quadratic's real-world applications and implications.