Problem 23 - College Algebra

Question

Solve the quadratic equation by using the square root property. $$ (2 x+1)^{2}=9 $$

Step-by-Step Solution

Answer

The solutions are \(x = 1\) and \(x = -2\).

Step 1: Understand the Equation

The given equation is \((2x + 1)^2 = 9\). To solve for \(x\) using the square root property, we need to isolate \((2x + 1)\) and then apply the square root to both sides.

Step 2: Apply the Square Root Property

The square root property states that if \(a^2 = b\), then \(a = \sqrt{b}\) or \(a = -\sqrt{b}\). Here, our \(a\) is \(2x + 1\) and \(b\) is 9. Applying the property, we have two equations: \(2x + 1 = 3\) and \(2x + 1 = -3\).

Step 3: Solve for x from the First Equation

Take the first equation \(2x + 1 = 3\) and solve for \(x\). Subtract 1 from both sides to get \(2x = 2\). Then, divide both sides by 2 to get \(x = 1\).

Step 4: Solve for x from the Second Equation

Take the second equation \(2x + 1 = -3\) and solve for \(x\). Subtract 1 from both sides to get \(2x = -4\). Then, divide both sides by 2 to get \(x = -2\).

Step 5: Verify Solutions

Substitute \(x = 1\) and \(x = -2\) back into the original equation to ensure they satisfy it. For \(x = 1\), \((2(1) + 1)^2 = 9\) holds true. For \(x = -2\), \((2(-2) + 1)^2 = 9\) also holds true. Both solutions are valid.

Key Concepts

Quadratic Equations

Quadratic equations are algebraic expressions of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients, and \(x\) represents the unknown variable. These equations are called 'quadratic' because the highest exponent of \(x\) is 2, stemming from the Latin word 'quadratus,' meaning square.
Quadratic equations have particular relevance in various fields such as physics, engineering, and finance because they can model real-world scenarios like projectile motion and area calculations. Solving quadratic equations often involves techniques such as factoring, completing the square, or using the quadratic formula. In cases where the equation is in a simple form like \((px + q)^2 = r\), the square root property offers a quick solution.
Recognizing quadratic equations in their various forms and being able to transform them into the standard form is crucial, as it allows you to select the most suitable method for solving them effectively.

Solving Equations

Solving equations involves finding the value of variables that make the equation true. In the context of quadratic equations, solving them means finding the value(s) of \(x\) that satisfy the equation perfectly.
When given an equation like \((2x + 1)^2 = 9\), the square root property is a handy tool. Rearranging and simplifying quadratic expressions helps isolate the term involving the variable, making it easier to apply solution methods like taking square roots.

• Start by isolating the quadratic term on one side of the equation.
• Apply algebraic methods to simplify the equation.
• Use direct algebraic properties such as the square root property, which simplifies expressions by binding them to positive and negative roots.

Solving quadratic equations can yield one or two real solutions, or possibly complex solutions, depending on the discriminant, indicating its versatility across problem-solving scenarios.

Algebraic Properties

Algebra involves various properties and operations that allow for systematic manipulation of equations to derive solutions. One critical algebraic property used in solving the original exercise is the square root property, which helps simplify quadratic expressions.
The square root property states that if \(a^2 = b\), then \(a = \sqrt{b}\) or \(a = -\sqrt{b}\). It can halve the complexity of an equation like \((2x + 1)^2 = 9\) by reducing it to two linear equations: \(2x + 1 = 3\) and \(2x + 1 = -3\).
This property leverages the concept of squaring and square roots, essential tools in algebra, allowing for direct calculations and highlighting the importance of understanding negative roots in quadratic equations. By using algebraic properties effectively, one can navigate through equations more fluidly and reach solutions more efficiently.