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Chapter 8: Problem 47

Which one of the two equations $$ (2 x+1)^{2}=5 \text { and } x^{2}+4 x=12 $$ is more suitable for solving by the square root property? By completing the square?

Short Answer

Expert verified
Equation (2x + 1)^2 = 5 is more suitable for solving by the Square Root Property. Equation x^2 + 4x = 12 is more suitable for Completing the Square.

Step by step solution

01

Identify the format of each equation

Examine the equations provided: Equation 1: (2x + 1)^2 = 5 Equation 2: x^2 + 4x = 12
02

Recognize the Square Root Property

The Square Root Property is suitable for equations that are easily written in the form A^2 = B. Equation 1 is already in this form: (2x + 1)^2 = 5, thus it can be solved using the Square Root Property.
03

Solve using Square Root Property

To solve (2x + 1)^2 = 5, apply the square root to both sides: (2x + 1) = 卤鈭�5 Then solve for x: 2x + 1 = 鈭�5 or 2x + 1 = -鈭�5 Thus, x = (鈭�5 - 1)/2 or x = (-鈭�5 - 1)/2
04

Complete the Square

To complete the square for the equation x^2 + 4x = 12, follow these steps: Rearrange the equation: x^2 + 4x - 12 = 0 Write it in the form x^2 + bx + c = 0: x^2 + 4x = 12
05

Completing the square

Add and subtract (b/2)^2 to the left hand side: x^2 + 4x + 4 = 12 + 4 This can be written as: (x + 2)^2 = 16 Now solve for x using the square root property: x+2 = 卤4 So, x = 2 or x = -2
06

Determine suitable methods

Equation 1, (2x + 1)^2 = 5, is more suitable for solving by the Square Root Property. Equation 2, x^2 + 4x = 12, is more suitable for Completing the Square.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Root Property
The Square Root Property is a method to solve quadratic equations and is very useful when the equation is in a specific form. Essentially, if we have an equation like \( A^2 = B \), we can solve for \( A \) by taking the square root of both sides.

For example, consider the equation \( (2x + 1)^2 = 5 \). It's already set up perfectly for this method.
  • First, take the square root of both sides: \( 2x + 1 = \pm \sqrt{5} \).
  • Next, solve for \( x \) by isolating it: \( 2x + 1 = \sqrt{5} \) or \( 2x + 1 = -\sqrt{5} \).
  • This gives us the solutions: \( x = (\sqrt{5} - 1)/2 \) and \( x = (-\sqrt{5} - 1)/2 \).
This property simplifies solving quadratics when they can be rewritten as something squared equals something else.
It bypasses the need for more complex methods, making it a powerful tool when applicable.
Completing the Square
Completing the square is a method to convert a quadratic equation into a perfect square trinomial. This is helpful for solving it or for rewriting the equation in a more usable form.

For instance, consider the quadratic equation \( x^2 + 4x = 12 \). To complete the square:
  • First, rearrange the equation: \( x^2 + 4x - 12 = 0 \).
  • Next, move the constant term to the other side: \( x^2 + 4x = 12 \).
  • Find \( (b/2)^2 \), where \( b \) is the coefficient of \( x \). Here, \( b = 4 \), so \( (4/2)^2 = 4 \).
  • Add and subtract this value to the left side to form a perfect square trinomial: \( x^2 + 4x + 4 = 12 + 4 \).
  • Now, rewrite the left side as a square of a binomial: \( (x + 2)^2 = 16 \).
  • Finally, solve for \( x \) using the Square Root Property: \( x+2 = \pm 4 \), thus \( x = 2 \) or \( x = -2 \).
This method is particularly useful when the quadratic equation does not easily factor or fit another straightforward method.
Quadratic Equations
Quadratic equations are polynomial equations of the form \( ax^2 + bx + c = 0 \) where \( a, b, \) and \( c \) are constants, and \( a eq 0 \). These equations often appear in various problem-solving scenarios, from physics to finance.

There are several methods to solve quadratic equations, such as:
  • Factoring
  • Using the Quadratic Formula
  • Completing the Square
  • Using the Square Root Property

Each method has its own advantages depending on the specific form of the quadratic equation. For instance:
  • The Quadratic Formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), is a universal method.
  • Factoring works well if the equation can be easily broken down into simpler binomials.
  • Completing the Square is handy for equations that resist factoring.
  • The Square Root Property is the fastest if the equation is already in the form \( A^2 = B \).
Understanding when and how to apply these methods will make solving quadratic equations much easier and more intuitive.

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