Problem 53 - Algebra 1: Concepts and Skills

Question

Solve the equation by factoring. Use a graphing calculator to check your solution if you wish. $$ 3 x^{2}-24 x+48=0 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 4\)

Step 1: Identify the Quadratic Equation

The given quadratic equation is \(3 x^{2}-24 x+48=0\). This can be written in the standard quadratic form \(ax^2 + bx + c = 0\), where \(a = 3\), \(b = -24\), and \(c = 48\).

Step 2: Factor the Quadratic Expression

To factor the quadratic expression, factor out the greatest common factor, which in this case is 3. This makes the equation easier to deal with: \[x^{2} - 8x + 16 = 0\] Now factor the remaining quadratic: \[(x-4)^2=0\]

Step 3: Solve for x

Finally, solve the equation for \(x\) by setting the factor to 0: \[x-4=0\] Solving this gives \[x=4\]

Key Concepts

Factoring

Factoring is a powerful method for solving quadratic equations. It involves expressing the quadratic equation in a product form, which then helps easily find its solutions. When you have an equation like \(3x^2 - 24x + 48 = 0\), the first step is to identify the greatest common factor (GCF). In this case, the GCF is 3. By dividing the entire equation by 3, we simplify it to \(x^2 - 8x + 16 = 0\).
Breaking down the simplified expression involves finding two numbers that multiply to give the constant term, 16, and add up to the linear coefficient, -8. By periodically practicing this, it starts to come naturally. Here, both conditions are satisfied by the number -4. Therefore, the expression can be written as a perfect square: \((x-4)^2 = 0\).
This form reveals that the equation has a repeated solution at \(x = 4\), indicating the line just touches the x-axis at this point. This makes factoring not only a solution-finding method but also a tool for visualizing quadratic graphs.

Quadratic Formula

The quadratic formula is another method to solve any quadratic equation, often used when factoring isn't straightforward. The formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), allows you to find solutions directly.
This formula comes especially handy for complex quadratics or when the factors aren't obvious integers. However, for our specific problem \(x^2 - 8x + 16 = 0\), establishing this into the formula can become an educational exercise. Insert \(a = 1\), \(b = -8\), and \(c = 16\) into the quadratic formula to check the consistency with the factor results.
The discriminant, \(b^2 - 4ac\), decides the nature of the roots: if it is positive, there are two distinct roots; zero means a double root; and negative indicates no real roots. Here, \((-8)^2 - 4(1)(16) = 0\) confirms there is one unique solution \(x = 4\), validating our earlier factoring approach.

Checking Solutions

Checking solutions can help reinforce the correct understanding and calculation of quadratic solutions. If you want to be certain about the solution \(x = 4\), plug it back into the original equation: \(3x^2 - 24x + 48 = 0\).
Substituting \(x = 4\) gives \(3(4)^2 - 24(4) + 48 = 0\). Calculate step by step:

• \(3 \times 16 = 48\)
• \(-24 \times 4 = -96\)
• Adding \(48 - 96 + 48 = 0\)

This checks out, confirming \(x = 4\) is indeed a correct solution.
Additionally, using a graphing calculator or an online graphing tool allows you to visualize the quadratic function. This assures the solution when the parabola touches or crosses the x-axis at x = 4. Such a graph confers confidence not just in finding numerical solutions but in understanding their real-world implications.