Problem 34 - Algebra and Trigonometry

Question

Solve equation by the square root property. $$ (2 x+8)^{2}=27 $$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{\sqrt{27} - 8}{2}\) and \(x = \frac{-\sqrt{27} - 8}{2}\)

Step 1: Isolate the quadratic term

Initially, the quadratic term, \((2x+8)^2\), is isolated. This isn't necessary for this problem, as the equation is already set equal to a constant, 27.

Step 2: Apply the square root property

Next, take the square root of both sides of the equation. This leads to: \[ 2x+8 = ± \sqrt{27} \]Remember to include both the positive and negative square roots as the square root property gives us two solutions.

Step 3: Solve for the variable

Finally, solve for the variable x. This can be done by initially subtracting 8 from both sides, leading to \(2x = ± \sqrt{27} - 8\). The last step is to divide by two, arriving at: \[ x = \frac{± \sqrt{27} - 8}{2} \]

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation that involves terms up to the second power. You can recognize it by its general formula:

• ax² + bx + c = 0

Here, 'a', 'b', and 'c' are constants, with 'a' not equal to zero. The quadratic equation in our exercise is slightly different. It's given in the form of

• (2x + 8)² = 27

This equation is already presented with the squared term isolated, which makes it a perfect candidate for using the square root property directly. Understanding these structures helps simplify the process of solving quadratic equations and clarifying what solutions to expect.

Solving Equations

Solving an equation means finding the value or values of the variable that make the equation true. For quadratic equations, this can involve a few different methods, such as factoring, completing the square, or using the quadratic formula. In some special cases, like our example, we utilize the square root property.
The process typically involves:

• Isolating the variable terms on one side of the equation.
• Eliminating constants and coefficients.
• Utilizing appropriate methods to find the variable values.

Solving equations requires careful manipulation to maintain equality on both sides, ensuring that the operations result in a mathematically valid solution.

Square Roots

Square roots are mathematical operations that determine what number, when multiplied by itself, produces the original number. The square root of a number 'n' is expressed as

• √n

When dealing with square roots in equations, particularly quadratic ones, it’s crucial to consider both the positive and negative roots.
This is because:

• Both square a positive and a negative number result in the same product: (√n)² = n and (-√n)² = n.

In our exercise,

• √27

results in ±√27, introducing two possible solutions to explore further. Understanding how square roots behave with numbers helps solve equations more effectively.

Variable Isolation

Variable isolation refers to the process of manipulating an equation to get the variable of interest by itself on one side of the equation. This step is essential to find the actual value(s) of the variable.
For instance, in our equation

• (2x + 8) = ±√27

we isolate 'x' by performing operations to eliminate other terms:

• Subtract 8 from both sides: 2x = ±√27 - 8
• Divide each side by 2: x = (±√27 - 8) / 2

Employing operations like addition, subtraction, multiplication, and division enables us to isolate variables systematically. This helps us solve for them more directly and clearly.