Problem 56 - Precalculus Mathematics for Calculus

Question

Solve the equation by completing the square. $$x^{2}-4 x+2=0$$

Step-by-Step Solution

Answer

The solutions are \(x = 2 + \sqrt{2}\) and \(x = 2 - \sqrt{2}\).

Step 1: Move constant to the other side

Start by isolating the constant term. Subtract 2 from both sides of the equation. \[x^2 - 4x = -2\]

Step 2: Prepare for completing the square

Take the coefficient of \(x\) which is \(-4\), divide it by 2 to get \(-2\), and then square it to get \(4\). Add this square to both sides of the equation.\[x^2 - 4x + 4 = -2 + 4\]

Step 3: Rewrite the left side as a square

The left-hand side of the equation is now a perfect square trinomial. Rewrite it as the square of a binomial.\[(x - 2)^2 = 2\]

Step 4: Solve for x by taking square roots

Take the square root of both sides of the equation to solve for \(x\). Remember to consider both the positive and negative roots.\[x - 2 = \pm \sqrt{2}\]

Step 5: Solve for x

Add 2 to both sides of the equation to solve for \(x\).\[x = 2 \pm \sqrt{2}\]

Key Concepts

Quadratic Equations

Quadratic equations are fundamental in algebra. They take the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). The equation represents a parabola when graphed. Most importantly, solving these equations can lead us to find the values of \(x\) that satisfy the equation. Depending on the quadratic, these equations can have two solutions, one solution, or no real solutions. Quadratics can be solved using various methods such as:

• Factoring
• Using the quadratic formula
• Completing the square (which is particularly useful when equations can't be easily factored)

Each method has its own advantages and application based on the equation given. Understanding the structure of a quadratic equation helps you decide the best method for solving it. For instance, if \(b\) is even and the trinomial doesn't factor neatly, completing the square may be the best choice.

Solving Equations

Solving equations is a key skill in algebra. It involves finding the value(s) of the variable that make the equation true. Let's break down the method of 'completing the square' with our example \(x^2 - 4x + 2 = 0\):To solve this:

• First, move the constant term to the other side to make the equation easier to work with. So, subtract \(2\) from both sides, yielding \(x^2 - 4x = -2\).
• Next, consider the term associated with \(x\). In this case, it is \(-4\). To complete the square, divide it by \(2\) (giving us \(-2\)), then square this result to get \(4\). Add \(4\) to both sides, making it a perfect square trinomial.
• Rewrite the left side as a binomial square: \((x - 2)^2 = 2\).
• Finally, solve for \(x\) by taking the square root of both sides, leading to \(x - 2 = \pm \sqrt{2}\). Add \(2\) to both sides to find \(x = 2 \pm \sqrt{2}\).

This effective strategy is helpful in dealing with quadratics that aren't easily factorable.

Algebra Concepts

Algebra is full of fascinating ideas that help you understand and manipulate mathematical expressions. One of these concepts is completing the square, a technique that not only aids in solving quadratic equations but also in deriving the quadratic formula and understanding the properties of parabolas.To complete the square means restructuring a quadratic equation so one side forms a perfect square trinomial. This is important because it transforms a complex quadratic into a simple binomial squared—a much easier form to solve. The perfect square trinomial ensures symmetry, making it clear how the variables and constants are related.Here's a quick recap of the process:

• Ensure the equation is in the form \(x^2 + bx = c\).
• Add \((b/2)^2\) to both sides to create a perfect square trinomial.
• Rewrite the trinomial as a squared binomial \((x + b/2)^2\).
• Solve for \(x\) by isolating the variable.

These steps show the power and versatility of algebraic manipulation, enhancing your problem-solving skills in mathematics.