Problem 19 - Algebra 2 and Trigonometry

Question

In \(9-26,\) solve each quadratic equation by completing the square. Express the answer in simplest radical form. $$ x^{2}-6 x+2=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = 3 + \sqrt{7}\) and \(x = 3 - \sqrt{7}\).

Step 1: Move the constant term to the other side

Start with the equation \(x^2 - 6x + 2 = 0\). Move the constant term, 2, to the other side of the equation by subtracting 2 from both sides, resulting in \(x^2 - 6x = -2\).

Step 2: Complete the square

To complete the square, take half of the coefficient of \(x\), which is -6, divide it by 2 to get -3, and then square it to get 9. Add and subtract 9 inside the equation in order to balance it: \(x^2 - 6x + 9 - 9 = -2\). This becomes: \((x - 3)^2 = 7\).

Step 3: Solve for x by taking the square root

Now solve \((x - 3)^2 = 7\) by taking the square root of both sides: \[ x - 3 = \pm \sqrt{7} \] This results from the property that \(\sqrt{a^2} = \pm a\).

Step 4: Solve for x

Add 3 to both sides to solve for \(x\): \[ x = 3 \pm \sqrt{7} \] This gives the two solutions: \(x = 3 + \sqrt{7}\) and \(x = 3 - \sqrt{7}\).

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of degree 2. It typically appears in the form \( ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(x\) represents an unknown variable. Quadratic equations are fundamental in algebra due to their applications and the unique properties of their solutions.

Solving quadratic equations can be done through various methods, including:

• Factoring, when the equation can be expressed as the product of two binomials
• Using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
• Completing the square, which restructures a quadratic into a perfect square trinomial

In this exercise, the method of completing the square is used to solve the quadratic equation. This method is helpful when factoring is difficult or impossible.

Simplest Radical Form

When solving quadratic equations, the solutions often involve square roots. The simplest radical form means expressing these square roots in a way that is as simplified as possible. This involves:

• Ensuring there are no perfect square factors under the radical sign
• Avoiding any fractions within the radical
• Writing the solution in a clear, concise manner

For example, after completing the square in the equation \((x - 3)^2 = 7\), we found \[ x = 3 \pm \sqrt{7} \]This expression is in its simplest radical form because 7 is already a prime number, and there are no further simplifications possible under the square root. Being able to recognize and express answers in simplest radical form ensures clarity and precision in mathematical solutions.

Square Root Property

The square root property is a valuable technique for solving equations where a variable is squared and isolated. It's based on the principle that if \(a^2 = b\), then \(a = \pm \sqrt{b}\). This allows us to eliminate the square by introducing positive and negative square roots into our solutions.

This property was pivotal in step 3 of solving the original exercise. After rewriting the equation as \((x - 3)^2 = 7\), we applied the square root property to solve:\[x - 3 = \pm \sqrt{7}\]Adding 3 to both sides results in the final solution:\[x = 3 \pm \sqrt{7}\]By using the square root property, we can effectively break down the squared expression and solve for the variable, leading to precise solutions. It is especially useful in problems involving quadratic equations that have been completed as squares.