Problem 59 - Intermediate Algebra

Question

Solve each quadratic equation using the method that seems most appropriate. $$ 4(2 x+1)^{2}-1=11 $$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{\sqrt{3} - 1}{2}\) and \(x = \frac{-\sqrt{3} - 1}{2}\).

Step 1: Simplify the Equation

Start by expanding the given equation if needed. The equation given is \(4(2x+1)^2 - 1 = 11\). First, add 1 to both sides of the equation to simplify it: \[4(2x+1)^2 - 1 + 1 = 11 + 1.\]Hence, we get:\[4(2x+1)^2 = 12.\]

Step 2: Divide Both Sides by 4

To isolate the squared term, divide both sides of the equation by 4:\[\frac{4(2x+1)^2}{4} = \frac{12}{4},\]which simplifies to:\[(2x+1)^2 = 3.\]

Step 3: Take the Square Root of Both Sides

To solve for \(2x+1\), take the square root of both sides:\[\sqrt{(2x+1)^2} = \sqrt{3}.\]This results in two possible solutions due to the square root:\[2x+1 = \sqrt{3}\quad \text{or} \quad 2x+1 = -\sqrt{3}.\]

Step 4: Solve for x

Solve each equation obtained from the previous step:**Case 1:**\(2x + 1 = \sqrt{3}\)Subtract 1 from both sides:\(2x = \sqrt{3} - 1.\)Divide by 2:\(x = \frac{\sqrt{3} - 1}{2}.\)**Case 2:**\(2x + 1 = -\sqrt{3}\)Subtract 1 from both sides:\(2x = -\sqrt{3} - 1.\)Divide by 2:\(x = \frac{-\sqrt{3} - 1}{2}.\)

Key Concepts

Solving Quadratics

Solving quadratic equations is a fundamental aspect of algebra that involves finding the values of the variable which make the equation true. Quadratic equations typically take the form \[ax^2 + bx + c = 0,\]where \(a\), \(b\), and \(c\) are constants. The solutions to these equations are the 'roots' or 'zeros' of the quadratic.
There are several techniques to solve quadratics, including:

• Factoring - useful when the equation can easily be broken down into simpler expressions.
• Quadratic Formula - applied to any quadratic equation and involves substituting the values into the formula \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.\]
• Completing the Square - modifies the equation to form a perfect square trinomial.
• Graphing - visually determining the roots for simple quadratics.
• Square Root Method - applicable when the quadratic can be simplified into a perfect square.

When choosing a method, it's beneficial to pick the one that aligns best with the structure of the equation you are solving. In the given exercise, the Square Root Method was chosen due to the simplified squared form.

Square Root Method

The Square Root Method is a straightforward procedure often applied to solve quadratics that take the form \((ax+b)^2 = c.\)This method is particularly effective when the quadratic is already or can be arranged into this squared format.
Here's how the method works:

• First, ensure the equation is in the form \((ax+b)^2 = c.\)
• Take the square root of both sides, remembering to consider both the positive and negative roots. This is due to the property that a squared term can yield both a positive and a negative solution when square rooted.
• Once the roots are extracted, solve the resulting linear equations for the variable to find the solution.

For example, in the exercise, after simplifying the equation to \((2x+1)^2 = 3,\)we take the square root to get \(2x+1 = \pm\sqrt{3}.\)This step gives two linear equations to solve: \(2x+1 = \sqrt{3}\) and \(2x+1 = -\sqrt{3}.\)Finally, each equation is solved for \(x\) to find the solutions.

Algebraic Techniques

Algebraic techniques are essential tools in mathematics, providing methods to isolate variables and simplify equations. When solving quadratic equations, these techniques become even more important to find accurate solutions.
In the context of quadratic equations, some key algebraic techniques include:

• Rearranging equations: This often involves moving terms from one side of the equation to the other to isolate the variable of interest. Algebraic symbols such as \(+, -, \times, \div\) are used strategically to maintain balance.
• Factorization: This technique involves expressing the equation as a product of its factors. If possible, the equation can be written as \((px+q)(rx+s) = 0,\)where the solutions can be easily derived from the factors.
• Substitution: Often used when dealing with one part of an equation separately before returning to the larger equation structure.
• Simplification: Techniques such as combining like terms, canceling common factors, or multiplying/dividing both sides by a constant to reduce complexity.

In the exercise, dividing both sides of \(4(2x+1)^2 = 12\)by 4 simplifies the equation to \((2x+1)^2 = 3,\)making subsequent steps more manageable. Mastering these techniques empowers students to tackle a range of mathematical challenges effectively.