Problem 34 - Introductory Algebra for College Students

Question

Solve each equation by the method of your choice. Simplify irrational solutions, if possible. $$(2 x+7)^{2}=25$$

Step-by-Step Solution

Answer

The solutions to the equation \((2 x+7)^{2}=25\) are \(x = -7, -1\)

Step 1: Expanded form

Expand the left-hand side of the equation \((2x+7)^2=25\). Expand as \((a+b)^2 = a^2+2ab+b^2\), which gives \(4x^2+28x+49=25\)

Step 2: Simplification

Simplify the equation into a standard quadratic form by subtracting 25 from both sides. This results in the final quadratic equation \(4x^2+28x+24=0\)

Step 3: Application of the Quadratic Formula

Now we can apply the quadratic formula where \(a=4\), \(b=28\), and \(c=24\). Which results in \(x = \frac{-28\pm\sqrt{28^2-4*4*24}}{2*4}\)

Step 4: Calculation of roots

Carrying out the calculations we get \(x = \frac{-28\pm\sqrt{784-384}}{8}\), leading to \(x = \frac{-28\pm\sqrt{400}}{8}\) and finally \(x=-7, -1\)

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to solve equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients. This formula is given by \(x = \frac{{-b \pm \sqrt{{b^2-4ac}}}}{{2a}}\). To apply this method, one should identify the coefficients corresponding to \(a\), \(b\), and \(c\), then substitute them into the formula. It's an effective technique when factoring is complex or not possible.

For example, in the provided exercise, after simplification, the equation \(4x^2+28x+24=0\) has coefficients \(a=4\), \(b=28\), and \(c=24\), which we plug into the quadratic formula. It facilitates finding the roots or solutions of the equation, which in this case are \(x=-7\) and \(x=-1\). It's important to simplify the expression under the square root, known as the discriminant, to make the calculations easier.

Factoring Quadratics

Factoring quadratics involves rewriting a quadratic equation as the product of two binomials. It is an alternative method to the quadratic formula and can be a quicker and easier approach, especially when dealing with simple or easily factorable expressions.

When factoring, look for two numbers that multiply to give the \(ac\) term (the product of the coefficients of \(x^2\) and the constant term) and add to give the \(b\) term (the coefficient of \(x\)). These numbers can be used to break up the middle term and factor by grouping. However, in our example with the equation \(4x^2+28x+24=0\), finding such numbers may not be straightforward, which hints at the utility of the quadratic formula in such scenarios.

Simplify Irrational Numbers

Irrational numbers often appear in quadratic equations when the discriminant \(b^2-4ac\) is not a perfect square. The process of simplifying involves expressing the square root in its simplest radical form or as a decimal if necessary. Ensuring that no perfect square factors remain under the radical is key.

In the context of the example, the discriminant simplifies to \(\sqrt{400}\), which is a perfect square and equals 20. Thus, no further simplification is needed, and we get rational solutions. However, when the discriminant isn't a perfect square, expressing the square root of prime factors or using a decimal approximation is usually the course taken to simplify the solution.

Algebraic Expressions

Algebraic expressions consist of variables, coefficients, and arithmetic operations. Manipulating these expressions is a fundamental part of solving algebraic equations, including quadratics. Steps in handling an algebraic expression typically include expansion, combining like terms, and simplification.

In our exercise, starting with \((2x+7)^2=25\), the binomial is expanded using the formula \((a+b)^2 = a^2+2ab+b^2\) to obtain an algebraic expression that can be simplified further. Mastery in manipulating algebraic expressions is crucial as it lays the groundwork for solving more complex equations.