Problem 50 - Algebra A Combined Function

Question

Solve each equation by completing the square. $$ 3 x^{2}-4 x=4 $$

Step-by-Step Solution

Answer

The solutions are \(x = 2\) and \(x = -\frac{2}{3}\).

Step 1: Move the Constant Term

First, move the constant term on the right-hand side to the left-hand side. Subtract 4 from both sides: \[ 3x^2 - 4x - 4 = 0 \].

Step 2: Divide by the Leading Coefficient

To make the coefficient of \(x^2\) equal to 1, divide every term in the equation by 3: \[ x^2 - \frac{4}{3}x - \frac{4}{3} = 0 \].

Step 3: Move the Constant Term Again

Move the constant term to the right-hand side by adding \(\frac{4}{3}\) to both sides: \[ x^2 - \frac{4}{3}x = \frac{4}{3} \].

Step 4: Complete the Square

To complete the square, take half of the coefficient of x (\(-\frac{4}{3}\)), square it, and add it to both sides. Half of \(-\frac{4}{3}\) is \(-\frac{2}{3}\). Squaring \(-\frac{2}{3}\) gives \(\frac{4}{9}\): \[ x^2 - \frac{4}{3}x + \frac{4}{9} = \frac{4}{3} + \frac{4}{9} \].

Step 5: Simplify and Express as a Square

Combine the terms on the right-hand side. The common denominator is 9, so: \[ \frac{4}{3} = \frac{12}{9} \rightarrow \frac{12}{9} + \frac{4}{9} = \frac{16}{9} \]. Write the left side as a perfect square: \[ \left(x - \frac{2}{3}\right)^2 = \frac{16}{9} \].

Step 6: Solve for x

Take the square root of both sides to solve for \(x\): \[ x - \frac{2}{3} = \pm \frac{4}{3} \]. This gives two solutions: \( x = \frac{2}{3} + \frac{4}{3} \) and \( x = \frac{2}{3} - \frac{4}{3}\).

Step 7: Simplify the Solutions

Simplify each solution: for \( x = \frac{2}{3} + \frac{4}{3} \), you get \( x = 2 \); for \( x = \frac{2}{3} - \frac{4}{3} \), you get \( x = -\frac{2}{3} \).

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of degree two, which generally take the form of \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable. These equations often result in parabolas when graphed on the coordinate plane.
Understanding quadratic equations is crucial as they frequently appear in various mathematical and real-world applications.

• Standard Form: As mentioned, the quadratic equation in its standard form is \( ax^2 + bx + c = 0 \).
• Roots: The solutions to the quadratic equation, also called roots, can be real or complex numbers. They are the x-values where the parabola intersects the x-axis.

Solving quadratic equations can involve different methods such as factoring, using the quadratic formula, or completing the square. In this exercise, we focused on completing the square to find the solutions for the equation \( 3x^2 - 4x = 4 \).

Algebraic Manipulation

Algebraic Manipulation is the skill of rearranging and simplifying algebraic expressions to make equations easier to work with. It’s like solving a puzzle, where the goal is to isolate the variable or make the equation more manageable.
The process of completing the square involves several algebraic manipulations:

• Moving Terms: Initially, you often move constants or known terms from one side to the other to prepare the equation for further simplification.
• Dividing by Coefficients: Dividing the entire equation by the leading coefficient (the coefficient of \( x^2 \)) helps simplify the equation and makes completing the square straightforward.
• Balancing the Equation: Whatever operation is done to one side of the equation must be performed on the other side to maintain equality.

These steps are fundamental when you work through solving quadratic equations by completing the square, as they prepare the equation to be easily rewritten in a squared binominal form.

Solving Equations

Solving equations can be thought of as a process of finding the values that satisfy a given equation. For quadratic equations, solving them means finding the values of \( x \) that make the equation true, essentially the points where the function crosses the x-axis on a graph.
Here is how solving was approached in the given exercise:

• Completing the Square: By transforming the left side into a perfect square trinomial, the equation becomes easier to solve using square roots.
• Taking Square Roots: Once the equation is in the form of \((x - d)^2 = e\), you can take the square root of both sides to find \( x \), which results in two possible solutions due to the \( \pm \) sign.
• Simplifying: After taking the square root and rearranging, simplify the solutions to their simplest numerical form.

Ultimately, this exercise concluded with two solutions: \( x = 2 \) and \( x = -\frac{2}{3} \), demonstrating how each methodical step leads to finding the correct roots.