Problem 21 - Algebra 2

Question

Solve each equation by using the Square Root Property. \(x^{2}-6 x+9=8\)

Step-by-Step Solution

Answer

The solutions are \(x = 3 + 2\sqrt{2}\) and \(x = 3 - 2\sqrt{2}\).

Step 1: Move Constant to the Right Side

The provided equation is \(x^2 - 6x + 9 = 8\). Start by moving the constant on the left side to the right side of the equation to isolate the \(x\) terms. This means subtracting 9 from both sides, giving \(x^2 - 6x + 9 - 9 = 8 - 9\). Simplifying this yields \(x^2 - 6x = -1\).

Step 2: Complete the Square

The expression \(x^2 - 6x\) needs to be completed to form a perfect square trinomial. Take half of the coefficient of \(x\), which is -6, \(\frac{-6}{2} = -3\), and then square it to get 9. Add and subtract 9 inside the equation to complete the square: \(x^2 - 6x + 9 = 8\). This simplifies to \((x - 3)^2 = 8\).

Step 3: Use the Square Root Property

To solve \((x - 3)^2 = 8\), apply the Square Root Property. This property implies that if \(a^2 = b\), then \(a = ±\sqrt{b}\). So, take the square root of both sides: \(x - 3 = ±\sqrt{8}\).

Step 4: Solve for x

Separate the equation into two possible solutions: \(x - 3 = \sqrt{8}\) and \(x - 3 = -\sqrt{8}\). Simplifying the square root gives \(\sqrt{8} = 2\sqrt{2}\). Solve both equations for \(x\):\(x = 3 + 2\sqrt{2}\) or \(x = 3 - 2\sqrt{2}\).

Key Concepts

Completing the Square

Completing the square is a method used to transform a quadratic expression into a perfect square trinomial. This simplifies the process of solving quadratic equations since a perfect square trinomial can be easily solved by applying the Square Root Property.
To complete the square, follow these steps:

• First, focus on the quadratic expression, which in our example is the left side of the equation: x^2 - 6x.
• Take half of the coefficient of the linear term, here it's -6. Half is {-6}/2 = -3.
• Square this result: (-3)^2 = 9.

Next, we manipulate the equation to reflect the perfect square trinomial. Add and subtract the square you found (9 in this case) to the equation. This results in (x^2 - 6x + 9 - 9 = -1 + 9), simplifying it to ((x - 3)^2 = 8).
By completing the square, we convert the quadratic into a form that's easily solvable, setting the stage for the application of the square root property.

Quadratic Equations

Quadratic equations are polynomial equations of degree two and have the form ax^2 + bx + c = 0. They appear in many different contexts in mathematics and the real world. Solving these types of equations often involves factorization, using the quadratic formula, or methods like completing the square.
In the given exercise, the original equation is a quadratic equation:

• x^2 - 6x + 9 = 8

It's essential to first rearrange it into the common form ax^2 + bx + c = 0.
By subtracting 8 from both sides, the equation becomes x^2 - 6x + 1 = 0, reminiscent of a classic quadratic equation format. This preparation allows us to explore varied methods of solution, such as completing the square in this instance.

Solving Equations

Solving equations is a fundamental aspect of algebra that involves finding the values of variables that make an equation true. For quadratic equations, there are multiple approaches to finding these values. Our focus here is on using the Square Root Property, a convenient method once the quadratic is expressed as a perfect square.
After completing the square in our example,

• (x - 3)^2 = 8

we apply the Square Root Property. This property states if a^2 = b, then a = ± (b).
Taking the square root of both sides yields two possible equations: x - 3 = (8) or x - 3 = - (8). Solving these gives x = 3 + 2 (2) and x = 3 - 2 (2), providing the solutions for the initial equation.
Mastering this method allows for straightforward problem-solving of similar quadratic equations, affirming the versatility of completing the square and applying square roots in algebraic tasks.