Problem 26 - College Algebra

Question

Solve the quadratic equation by completing the square. Show each step. $$ 2 x^{2}-8 x-5=0 $$

Step-by-Step Solution

Answer

The solutions are \( x = 2 + \sqrt{\frac{13}{2}} \) and \( x = 2 - \sqrt{\frac{13}{2}} \).

Step 1: Write the Equation in Standard Form

Start with the given quadratic equation: \[ 2x^2 - 8x - 5 = 0 \] The equation is already in its standard form \( ax^2 + bx + c = 0 \), where \( a = 2 \), \( b = -8 \), and \( c = -5 \).

Step 2: Divide by the Leading Coefficient

To complete the square, we need the coefficient of \( x^2 \) to be 1. So, divide the entire equation by 2:\[ x^2 - 4x - \frac{5}{2} = 0 \]

Step 3: Move the Constant Term to the Other Side

Add \( \frac{5}{2} \) to both sides of the equation to isolate the quadratic and linear terms:\[ x^2 - 4x = \frac{5}{2} \]

Step 4: Complete the Square

Find the number that completes the square. Take half of the coefficient of \( x \) (which is -4), square it, and add that square to both sides of the equation:\[ x^2 - 4x + 4 = \frac{5}{2} + 4 \]Here, half of -4 is -2 and \(-2^2 = 4\). So, we add 4 to both sides:\[ x^2 - 4x + 4 = \frac{5}{2} + 4 \]

Step 5: Simplify the Right Side

Calculate the right side:\[ \frac{5}{2} + 4 = \frac{5}{2} + \frac{8}{2} = \frac{13}{2} \]The equation becomes:\[ (x-2)^2 = \frac{13}{2} \]

Step 6: Solve for x by Taking the Square Root

Take the square root of both sides to solve for \( x \):\[ x - 2 = \pm \sqrt{\frac{13}{2}} \]So,\[ x = 2 \pm \sqrt{\frac{13}{2}} \]

Step 7: Simplify the Expression

Distribute the \( \pm \) to get two solutions:\[ x = 2 + \sqrt{\frac{13}{2}} \text{ or } x = 2 - \sqrt{\frac{13}{2}} \]

Key Concepts

Quadratic Equations

Quadratic equations are a type of polynomial equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). These equations are fundamental in algebra, as they can describe many real-world phenomena from physics to economics. In solving quadratic equations, there are multiple methods: factoring, using the quadratic formula, and completing the square, which is the focus here.

Completing the square involves manipulating the equation to form a perfect square trinomial, which simplifies solving considerably. This method can be particularly useful when the quadratic equation cannot be easily factored. It is also an essential step in deriving the formula for the vertex of a parabola, a high school algebra concept. Understanding each component of the quadratic equation is crucial for using this method effectively.

• \( a \) is the coefficient of \( x^2 \) and determines the equation's leading term.
• \( b \) is the coefficient of \( x \), influencing the equation's linear component.
• \( c \) is the constant term, contributing to the equation's constant.

Solving Equations

Solving equations is a foundational skill in algebra. It involves finding all possible values of the variable that satisfy the equation. For quadratic equations, solving can be broken down into understanding and manipulating expressions to isolate the variable. The equation \( 2x^2 - 8x - 5 = 0 \) can be solved using completing the square which provides a structured approach to solve such quadratic equations.

Here's a high-level look at the process:

• First, transform the quadratic into one where \( x^2 \) has a coefficient of 1. This is achieved by dividing all terms by the leading coefficient if it isn't 1 already.
• Next, re-arrange the equation so that the constants are on the other side.
• Then, generate a perfect square trinomial by finding a value that completes the square and add it to both sides.

Once these steps are completed, the equation can be simplified further to find the solution using square roots. Solving quadratic equations using this method allows students to deepen their understanding of algebraic manipulation and why different algebraic techniques work.

Algebraic Manipulation

Algebraic manipulation is a process of rearranging and simplifying equations to find solutions or expressions in a workable form. This process requires a solid grasp of algebraic properties and operations, including understanding equality and balance within equations.

In completing the square, algebraic manipulation plays a central role. The steps involved include:

• Dividing the entire equation by the leading coefficient to simplify the equation, particularly useful if the quadratic's leading term is not already one (as seen in the problem \( 2x^2 - 8x - 5 = 0 \)).
• Moving terms across the equation to isolate variables of interest. This involves a clear understanding of transposition and balance within the equation.
• Adding a square term to both sides to transform the quadratic equation into a perfect square trinomial, a key act of manipulation, which ensures that the integrity of the equation is maintained while facilitating the path to solution.

This kind of manipulation allows the quadratic equation to be restructured into a simpler form \((x - h)^2 = k\), ultimately making it easier to solve using square roots. Students must practice algebraic manipulation to skillfully navigate complex algebra operations efficiently.