Problem 57 - Algebra 2

Question

Solve each equation by using the Square Root Property. \(x^{2}-8 x+16=7\)

Step-by-Step Solution

Answer

The solutions are \(x = 7\) and \(x = 1\).

Step 1: Identify the Quadratic Equation

The given equation is \(x^{2} - 8x + 16 = 7\). This is a quadratic equation which we can simplify to use the Square Root Property.

Step 2: Simplify the Equation

First, we need to move all the terms to one side of the equation: \(x^2 - 8x + 16 - 7 = 0\). This simplifies to \(x^2 - 8x + 9 = 0\).

Step 3: Reorganize into a Perfect Square

Recognize the left side of the equation as a perfect square: \((x - 4)^2\). Hence, rewrite the equation as \((x - 4)^2 = 9\).

Step 4: Use the Square Root Property

Apply the Square Root Property which states if \(a^2 = b\), then \(a = \pm \sqrt{b}\). Therefore, \(x - 4 = \pm 3\).

Step 5: Solve for x

Solve the two resulting equations. First, \(x - 4 = 3\) gives \(x = 7\). Second, \(x - 4 = -3\) gives \(x = 1\).

Key Concepts

Quadratic Equation

A quadratic equation is a type of mathematical equation where the highest exponent of the variable is 2. This means it usually takes the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. In our original example, the quadratic equation is \(x^2 - 8x + 16 = 7\). To handle quadratic equations effectively, we often want to move all terms to one side so that it equals zero.

• This helps in setting up the equation for factoring or applying other solving methods like the Square Root Property.
• Quadratic equations can have two solutions because they graph as a parabola, and these solutions are the points where the parabola intersects the x-axis.

To realign the equation to standard form, subtract 7 from both sides, giving us \(x^2 - 8x + 9 = 0\). This setup facilitates recognizing the equation's structure and identifying potential perfect squares.

Perfect Square

A perfect square in algebra is an expression that can be written as the square of a binomial. For example, \((x - 4)^2\) is a perfect square because it expands to \(x^2 - 8x + 16\). Recognizing a perfect square is crucial when using methods like the Square Root Property.

• Perfect squares are helpful when solving quadratic equations as they simplify expressions, revealing solutions more easily.
• By factoring the quadratic \(x^2 - 8x + 9\), we reorganize it into \((x - 4)^2 = 9\). Here, the perfect square makes it clear that the equation is symmetric around \(x = 4\).

Identifying this symmetry efficiently turns the equation into one that can be solved simply by applying the square root to both sides, streamlining the path to the equation’s solutions.

Simplifying Equations

Simplifying equations is a helpful skill in algebra. It involves rearranging and reducing the equation to a format that’s easier to solve. In our example, we began with \(x^2 - 8x + 16 = 7\) and simplified it to \((x - 4)^2 = 9\).

• Simplifying an equation often helps in reducing computation complexity, leading to quicker solutions.
• For quadratic equations of the form \(ax^2 + bx + c\), identifying the perfect square helps in transforming and simplifying them.

Once the equation is simplified to \((x - 4)^2 = 9\), solving becomes straightforward using the Square Root Property. This process demonstrates the power of simplification in making complex problems more manageable, paving the way to easily find the roots \(x = 7\) and \(x = 1\).