Problem 7 - Introductory Algebra for College Students

Question

Solve each equation using the quadratic formula. Simplify irrational solutions, if possible. $$x^{2}+4 x-7=0$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -2 + \sqrt{11}\) and \(x = -2 - \sqrt{11}\).

Step 1: Identify 'a', 'b', and 'c'

From the quadratic equation \(x^{2} + 4x - 7 = 0\), we identify 'a', 'b', and 'c' as follows: \(a = 1\), \(b = 4\), and \(c = -7\) respectively.

Step 2: Substitute in the quadratic formula

We substitute the values of 'a', 'b', and 'c' into the quadratic formula \(x = \frac{-b \pm \sqrt{b^{2}- 4ac}}{2a}\), and simplify under the square root where necessary. This gives us \(x = \frac{-4 \pm \sqrt{4^{2}- 4 \times 1 \times -7}}{2 \times 1}\) which is \(x = \frac{-4 \pm \sqrt{16 + 28}}{2}\), and that further simplifies to \(x = \frac{-4 \pm \sqrt{44}}{2}\).

Step 3: Simplify further

Notice that 44 is \(4 \times 11\), and because 4 is a perfect square, we can simplify under the square root. This gives us \(x = \frac{-4 \pm 2\sqrt{11}}{2}\), which further simplifies to \(x = -2 \pm \sqrt{11}\) after dividing both terms in the numerator by 2.

Key Concepts

Simplifying Irrational Solutions

Simplifying irrational solutions is an essential skill when working with quadratic equations, particularly after using the quadratic formula. When a solution is irrational, it contains a square root that doesn’t simplify to a whole number. Let's see how we can simplify these kinds of solutions.

After applying the quadratic formula, we sometimes end up with a square root in the solution. For instance, in our equation, we reached the expression \(x = \frac{-4 \pm \sqrt{44}}{2}\). Notice that 44 can be broken down into \(4 \times 11\).

• The number 4 is a perfect square, it simplifies to 2.
• This allows \(\sqrt{44}\) to be rewritten as \(\sqrt{4 \times 11} = \sqrt{4} \times \sqrt{11} = 2\sqrt{11}\).

Simplifying further, we divide each part of the numerator by the common factor 2, giving us \(x = -2 \pm \sqrt{11}\). Simplifying irrational solutions in such a manner makes the final answer much neater and easier to work with.

Quadratic Equations

Quadratic equations are a fundamental part of algebra and appear in many different forms. They are polynomial equations of degree two, generally expressed as \(ax^2 + bx + c = 0\), where:\

• \(a\), \(b\), and \(c\) are constants.
  • The term \(ax^2\) is the quadratic term.
  • \(bx\) is the linear term.
  • \(c\) is the constant term.

The quadratic formula, \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\), provides a straightforward way to find solutions. It works for any quadratic equation, offering solutions even when they aren't integers or easy to factor.

In our example, \(x^2 + 4x - 7 = 0\), we identified \(a = 1\), \(b = 4\), and \(c = -7\), and used these values in the quadratic formula to find the roots of the equation.

Algebra Fundamentals

Algebra fundamentals encompass the basic concepts and techniques used to manipulate algebraic expressions and equations. Understanding these basics is crucial when dealing with any problem involving equations like the quadratic one.

To start with, identifying coefficients \(a\), \(b\), and \(c\) in a quadratic equation is a fundamental skill. This step is essential because these values are used in the quadratic formula. In rearranging terms or simplifying expressions, maintaining proper arithmetic is crucial.

• For example, correctly simplifying \(\sqrt{44} = \sqrt{4 \times 11} = 2\sqrt{11}\).
• Operating with fractions: dividing terms by common numbers, such as reducing \(\frac{-4 \pm 2\sqrt{11}}{2}\) to \(-2 \pm \sqrt{11}\).

These techniques provide a strong foundation that helps when progressing to more complex algebraic concepts. Mastery of these fundamentals enables you to approach more challenging problems with confidence.