Problem 18 - Algebra and Trigonometry

Question

Solve the equation by completing the square. \(x^{2}-5 x+1=0\)

Step-by-Step Solution

Answer

\(x = 2.5 \pm \sqrt{5.25}\)

Step 1: Move the Constant Term

To complete the square, first move the constant term to the right side of the equation. Start with the equation: \[x^2 - 5x + 1 = 0\]Subtract 1 from both sides:\[x^2 - 5x = -1\]

Step 2: Identify the Coefficient and Halve It

Identify the coefficient of the linear term (i.e., \(-5\)), halve it, and then square it. Calculate:\[-\frac{5}{2} = -2.5,\]Squaring it gives:\[(-2.5)^2 = 6.25\]

Step 3: Add and Subtract the Square

Add and subtract \(6.25\) to/from the left-hand side to maintain equality:\[x^2 - 5x + 6.25 - 6.25 = -1\]Rearrange it as:\[x^2 - 5x + 6.25 = 5.25\]

Step 4: Write as a Perfect Square Trinomial

Write the left-hand side of the equation as a perfect square trinomial:\[(x - 2.5)^2 = 5.25\]

Step 5: Solve for \(x\)

Take the square root of both sides to solve for \(x\):\[x - 2.5 = \pm \sqrt{5.25}\]Thus, solving for \(x\):\[x = 2.5 \pm \sqrt{5.25}\]

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of the second degree. They are generally written in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). These equations have important applications in various fields such as physics, engineering, and economics.

Here's a quick breakdown of the components:

• \(a\) is the coefficient of \(x^2\) and should be non-zero.
• \(b\) is the coefficient of \(x\). It affects the axis of symmetry of the equation's graph.
• \(c\) is the constant term, which alters the equation's y-intercept on a graph.

Quadratic equations have solutions known as "roots." These roots can be real or complex numbers. They represent the x-values where the graph of the equation intersects the x-axis. Several methods can be used to find the roots of a quadratic equation, including factoring, the quadratic formula, and completing the square.

In the context of the original problem \(x^2 - 5x + 1 = 0\), we aim to solve it by completing the square, a method that involves creating a perfect square trinomial to simplify finding the roots.

Perfect Square Trinomial

A perfect square trinomial is a quadratic expression that can be written as the square of a binomial. It takes the form \((x + d)^2 = x^2 + 2dx + d^2\) or \((x - d)^2 = x^2 - 2dx + d^2\). This structure makes them particularly easy to solve because we can directly take the square root of both sides.

The process of creating a perfect square trinomial involves completing the square. Here's how you do it:

• Start with your quadratic expression, usually in the form \(x^2 + bx\).
• Identify \(b\), the coefficient of the linear term \(x\).
• Take half of \(b\) and square it: \(\left(\frac{b}{2}\right)^2\).
• Add this square to your quadratic expression to create the perfect square trinomial.

In our specific problem, we had a linear coefficient of \(-5\). Halving it gave us \(-2.5\), and squaring this value provided \(6.25\). Adding and subtracting \(6.25\) allowed us to rewrite the equation as a perfect square trinomial \((x - 2.5)^2\). This step is crucial because it transforms the equation into a form that's easier to solve.

Solving Equations

After converting an equation into a perfect square trinomial, solving it becomes a straightforward process. Here's what involves solving these equations using the square root method:

1. **Take the square root of both sides:** Once you have a perfect square, take the square root of both sides of the equation. This will give you an expression without the square. In our example: \((x - 2.5)^2 = 5.25\) becomes \(x - 2.5 = \pm \sqrt{5.25}\).

2. **Solve for \(x\):** Add or subtract the constant term from both sides to isolate \(x\). This gives you the roots of the original quadratic equation. For the equation \(x - 2.5 = \pm \sqrt{5.25}\), adding \(2.5\) to both sides results in:

• \(x = 2.5 + \sqrt{5.25}\)
• \(x = 2.5 - \sqrt{5.25}\)

Solving quadratic equations through completing the square is a reliable technique. It not only helps in finding the roots but also increases your understanding of the quadratic expression structure. This method can also be applied to equations that cannot be easily factored, making it a versatile tool in algebra.