Problem 66 - A Graphical Approach to Precalculus with Limits

Question

Solve each quadratic equation by completing the square. $$x(x-2)=4$$

Step-by-Step Solution

Answer

The solutions are \( x = 1 \pm \sqrt{5} \).

Step 1: Expand the Equation

Start by expanding the expression on the left side of the equation: \[ x(x - 2) = 4 \]Expanding, we have:\[ x^2 - 2x = 4 \]

Step 2: Move the Constant Term

Rearrange the equation such that we can complete the square by moving the constant term to the right side:\[ x^2 - 2x = 4 \]

Step 3: Complete the Square

To complete the square, take half of the coefficient of \( x \) and square it. The coefficient of \( x \) is \(-2\). Half of \(-2\) is \(-1\), and \((-1)^2 = 1\). Add and subtract this square on the left side:\[ x^2 - 2x + 1 - 1 = 4 \]This becomes:\[ (x - 1)^2 - 1 = 4 \]

Step 4: Solve for the Perfect Square

Add 1 to both sides to isolate the perfect square:\[ (x - 1)^2 = 5 \]

Step 5: Take the Square Root of Both Sides

Take the square root of both sides of the equation to solve for \( x \):\[ x - 1 = \pm \sqrt{5} \]

Step 6: Solve for x

Add 1 to both sides to find the values for \( x \):\[ x = 1 \pm \sqrt{5} \]This gives the solutions:\[ x = 1 + \sqrt{5} \] and \[ x = 1 - \sqrt{5} \]

Key Concepts

Quadratic Equations

Quadratic equations form a fundamental part of algebra, involving expressions of the form \( ax^2 + bx + c = 0 \). The equation represents a parabola when plotted on a graph. Understanding quadratic equations is essential because they appear in various mathematical and real-world contexts. The solutions to quadratic equations, also called roots, can be found using several methods like factoring, using the quadratic formula, or completing the square.

• The general form consists of a variable squared \((x^2)\), a linear term \((bx)\), and a constant \((c)\).
• These equations are always solved for \(x\) where the highest exponent is 2.

In our exercise, we started with the quadratic equation derived from multiplying through the expression \( x(x - 2) = 4 \). After expansion, we worked with \( x^2 - 2x = 4 \), showcasing a typical quadratic form \( ax^2 + bx = c \) with \( c \) temporarily on the other side.

Algebraic Manipulation

Algebraic manipulation involves transforming equations and expressions to achieve a specific form. This skill is crucial in solving equations, as it allows the user to isolate variables, simplify expressions, and apply algebraic methods effectively. Completing the square is one such technique involving algebraic manipulation.

• Isolate one variable to simplify solving.
• Move constants and like terms to different sides to prepare for completing the square.

In our case, manipulating \( x^2 - 2x = 4 \) involved moving the constant term to facilitate completing the square. Rearranging as \( x^2 - 2x + 1 = 5 \) was an essential step, preparing the equation for isolation and solution of \( x \). This manipulation set the stage for "completing the square," aiding in solving the quadratic equation accurately.

Perfect Square Trinomial

A perfect square trinomial is an expression that is the square of a binomial, i.e., \((a + b)^2 = a^2 + 2ab + b^2\). Recognizing or creating a perfect square trinomial helps to simplify solving some quadratic equations.

• Add and subtract the square of half the linear coefficient \(b\) to complete the square on one side.
• Convert expressions into the form \((a + b)^2 = c\) to use square root methods for simpler solutions.

In our exercise, identifying the square of \(-1\) as 1 was key to creating \( (x - 1)^2 \). By rewriting \( x^2 - 2x \) as \( (x - 1)^2 - 1 \), we transformed the equation into a format that's easier to solve. This method helped isolate \( x \), allowing for straightforward solution by taking square roots. Recognizing how to complete the square can offer a powerful strategy for solving a wide range of quadratic problems.