Problem 17 - Algebra 2

Question

Solve each equation by using the Square Root Property. \(x^{2}-10 x+25=49\)

Step-by-Step Solution

Answer

The solutions are \(x = 12\) and \(x = -2\).

Step 1: Recognize Perfect Square

First, identify that the left side of the equation \(x^2 - 10x + 25\) is a perfect square trinomial. It can be rewritten as \((x-5)^2\). This simplifies our equation to \((x-5)^2 = 49\).

Step 2: Apply Square Root Property

To solve \((x-5)^2 = 49\), apply the square root property by taking the square root of both sides: \(x-5 = \pm 7\). This gives us two possible equations to solve: \(x-5 = 7\) and \(x-5 = -7\).

Step 3: Solve Each Equation

Solve \(x-5 = 7\) by adding 5 to both sides to get \(x = 12\). Similarly, solve \(x-5 = -7\) by adding 5 to both sides to get \(x = -2\).

Step 4: State the Solutions

The solutions to the equation \((x-5)^2 = 49\) are \(x = 12\) and \(x = -2\).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic expression. Recognizing it can simplify solving equations. It happens when you can express a polynomial as the square of a binomial. In other words, if you have something like \(a^2 + 2ab + b^2\), it can be rewritten using \((a+b)^2\).
In the exercise given, \(x^2 - 10x + 25\) is our perfect square trinomial. Here, let's identify the components:

• The first term, \(x^2\), is the square of \(x\).
• The second term, \(-10x\), is twice the product of \(x\) and \(5\).
• Finally, \(25\) is the square of \(5\).

So, we can simplify \(x^2 - 10x + 25\) into \((x-5)^2\). This is crucial, as it turns a complicated expression into a simpler form, making equation solving easier.

Equation Solving

Equation solving involves finding the variable's value that satisfies the equation. In this exercise, once the trinomial is expressed as \((x - 5)^2\), solving becomes straightforward using the Square Root Property.
The Square Root Property states that if \(a^2 = b\), then \(a = \pm \sqrt{b}\). Applying this property allows us to solve \((x-5)^2 = 49\). By taking the square root of both sides, we obtain two separate equations due to the \(\pm\):

• \(x - 5 = 7\)
• \(x - 5 = -7\)

By solving these simple linear equations, we find the values of \(x\) that make the equation true, specifically \(x = 12\) and \(x = -2\). This process shows how breaking down a problem into known forms can simplify equation solving.

Quadratic Equations

Quadratic equations are those that involve terms up to the second power of the variable. They take a general form \(ax^2 + bx + c = 0\). Solving these can sometimes be challenging, but identifying certain properties, like perfect square trinomials, can help.
In our example, the equation \(x^2 - 10x + 25 = 49\) is adjusted to the standard form \((x-5)^2 - 49 = 0\). Recognizing it as a perfect square trinomial saves time because it condenses the process to a simple square root calculation.
Quadratic equations can often be solved using methods such as:

• Factoring
• Completing the square
• Using the quadratic formula
• Graphical solutions

Each method has its context and advantages. Choosing wisely based on the equation's form is key to efficient solving. In this particular exercise, using the Square Root Property after recognizing the perfect square was the best route, illustrating the power of understanding underlying concepts.