Problem 50 - Elementary Algebra

Question

Solve by completing the square. $$2 x_{2}-4 x-3=0$$

Step-by-Step Solution

Answer

The solutions are \(x = 1 + \sqrt{\frac{5}{2}}\) and \(x = 1 - \sqrt{\frac{5}{2}}\).

Step 1: Move the Constant

To complete the square, we first need to move the constant term to the other side of the equation. Start with the equation:\[ 2x^2 - 4x - 3 = 0 \]Add 3 to both sides:\[ 2x^2 - 4x = 3 \]

Step 2: Simplify the Quadratic Term Coefficient

Divide the entire equation by 2 to make the coefficient of the \(x^2\) term equal to 1:\[ x^2 - 2x = \frac{3}{2} \]

Step 3: Complete the Square

Take half of the coefficient of \(x\), square it, and add it to both sides. The coefficient of \(x\) is -2. Half of -2 is -1, and -1 squared is 1:Add 1 to both sides:\[ x^2 - 2x + 1 = \frac{3}{2} + 1 \]Simplify the right side:\[ x^2 - 2x + 1 = \frac{5}{2} \]

Step 4: Express as a Perfect Square

Re-write the left side of the equation as a squared term:\[ (x - 1)^2 = \frac{5}{2} \]

Step 5: Solve for x

Take the square root of both sides:\[ x - 1 = \pm \sqrt{\frac{5}{2}} \]Add 1 to both sides to solve for x:\[ x = 1 \pm \sqrt{\frac{5}{2}} \]

Step 6: Express the Solutions

The solutions are:\[ x = 1 + \sqrt{\frac{5}{2}} \; \text{or} \; x = 1 - \sqrt{\frac{5}{2}} \]

Key Concepts

Completing the Square

Completing the square is a method used to solve quadratic equations. The main idea is to transform a quadratic equation into a perfect square trinomial. This is especially useful because perfect square trinomials can be factored into the square of a binomial, which is easier to solve. Let's break it down.

At its core, completing the square involves several critical steps:

• First, ensure the coefficient of the quadratic term (\( x^2 \)) is 1. If it isn't, divide the entire equation by the coefficient.
• Next, move the constant term to the opposite side of the equation. This will set up the equation to be completed.
• Then, take half of the linear coefficient (the term in front of\( x \) ), square it, and add this value to both sides of the equation. This creates a perfect square trinomial on one side.
• Finally, express this trinomial as the square of a binomial.

These steps help in rewriting the equation in a form that can be easily solved using algebraic techniques.

Solving Quadratic Equations

Solving quadratic equations is an essential skill in algebra. These equations have the form \( ax^2 + bx + c = 0 \), where\( a \), \( b \), and \( c \) are constants. Different methods can be used, such as factoring, using the quadratic formula, or completing the square.

When choosing a solving method, it's essential to identify the most efficient one:

• Factoring works well when the quadratic can easily be rewritten as a product of binomials.
• Quadratic formula is universal but can be cumbersome with complex numbers or decimals.
• Completing the square is highly effective when the quadratic term's coefficient is one. It also helps understand the vertex form of a parabola.

Each strategy has its uses and choosing the right one depends on the equation you are working with.

Step-by-step Algebra Solutions

By approaching algebra problems step-by-step, you can gain clarity and precision in solving equations. Let's see how to solve a quadratic by completing the square using a systematic approach.

Here’s a breakdown of the steps using the equation \( 2x^2 - 4x - 3 = 0 \):1. Move the Constant: Begin by isolating the \( x \) terms by shifting the constant to the other side. This simplifies the setup. Here, we add 3, leading to \( 2x^2 - 4x = 3 \).2. Adjust the Quadratic Coefficient: Divide the whole equation by 2 to normalize the quadratic term: \( x^2 - 2x = \frac{3}{2} \).3. Complete the Square: Half of the \(-2\) is \(-1\) and squaring it gives 1. Add this to both sides, forming\( x^2 - 2x + 1 = \frac{5}{2} \).4. Reform the Equation: Express as a square: \( (x - 1)^2 = \frac{5}{2} \).5. Solve for \( x \): Take the square root: \( x - 1 = \pm \sqrt{\frac{5}{2}} \), and then add 1 to both sides to find \( x = 1 \pm \sqrt{\frac{5}{2}} \).By following these clear steps, understanding and solving complex algebraic problems becomes more manageable.