Problem 26 - Introductory Algebra for College Students

Question

Solve quadratic equation by completing the square. \(x^{2}=5 x-3\)

Step-by-Step Solution

Answer

The solutions to the given quadratic equation are \(x = 2.5 + \sqrt{9.25}\) and \(x = 2.5 - \sqrt{9.25}\).

Step 1: Rearrange the equation

First, the original equation \(x^{2} - 5x + 3 = 0\) needs to be rearranged to the form \(x^2 - 5x = 3\).

Step 2: Complete the square

The quadratic term in the equation needs to be completed by adding the square of half the coefficient of 'x'. Therefore, add \(\left(\frac{-5}{2}\right)^2 = 6.25\) to both sides to get \((x - 2.5)^2 = 9.25\).

Step 3: Find the value of 'x'

Finally, to find 'x', take the square root of both sides. Since the square root has both a positive and negative value, there are two possible solutions. \nTherefore, the solutions would be \(x = 2.5 + \sqrt{9.25}\) and \(x = 2.5 - \sqrt{9.25}\).

Key Concepts

Completing the Square

Completing the square is a method used to solve quadratic equations by transforming a quadratic expression into a perfect square trinomial. This method is beneficial when a quadratic equation doesn't factor easily. The goal is to rewrite the equation in such a way that it becomes easier to solve for the variable.

To complete the square, follow these steps:

• Ensure the quadratic equation is in the form \[x^2 + bx + c = 0\]
• Move the constant term to the other side: \[x^2 + bx = -c\]
• Calculate and add \[\left(\frac{b}{2}\right)^2\] to both sides to form a perfect square trinomial.
• This changes the left side of the equation to \[(x + \frac{b}{2})^2\]

By completing the square, the equation \[x^2 - 5x + 3 = 0\] becomes \[(x - 2.5)^2\], which is easier to solve by taking square roots.

This method is especially useful for deriving other forms of the quadratic formula and is a fundamental concept in algebra.

Quadratic Formula

The quadratic formula is a universal tool for solving any quadratic equation of the form \[ax^2 + bx + c = 0\]. This method doesn't rely on factoring and can solve any type of quadratic equation when the equation is difficult to factor.

The formula is: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
- \(b^2 - 4ac \) is the discriminant, determining the nature of the roots.

Here's how it works:

• Identify values of \(a\), \(b\), and \(c\) from the quadratic equation.
• Plug these values into the formula.
• Calculate the discriminant \(b^2 - 4ac\).
• Evaluate the solutions which could be real or complex.

When using the quadratic formula for \[x^2 - 5x + 3 = 0\], calculations might be complex, but the formula provides a straightforward solution for any quadratic situation.

It complements other solving methods like completing the square, particularly when manual calculations are complex.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations or expressions to make them easier to work with or solve. It's a fundamental technique in solving equations, especially quadratics.

Key techniques include:

• Rearranging terms to isolate the variable.
• Combining like terms to simplify expressions.
• Factoring to express an equation in simpler terms.

In this specific problem: \[x^2 - 5x + 3 = 0\], we start by rearranging it to begin the completing the square process: \[x^2 - 5x = -3\].

This rearrangement is essential for applying methods like completing the square or the quadratic formula.

Algebraic manipulation lays the groundwork for solving equations systematically. Mastering these techniques improves understanding and effectiveness in dealing with various algebraic problems.