Problem 4 - Introductory Algebra for College Students

Question

Solve each equation using the quadratic formula. Simplify irrational solutions, if possible. $$x^{2}+5 x+2=0$$

Step-by-Step Solution

Answer

The solutions of the quadratic equation \(x^{2} + 5x + 2 = 0\) are \(x = \frac{{-5 + \sqrt{17}}}{2}\) and \(x = \frac{{-5 - \sqrt{17}}}{2}\).

Step 1: Identify A, B and C

Coefficients from the equation \( x^{2} + 5x + 2 = 0 \) are A=1, B=5, C=2.

Step 2: Substitute in the Quadratic Formula

Substitute A=1, B=5, and C=2 into the quadratic formula \(x = \frac{{-B \pm \sqrt{{B^{2} - 4AC}}}}{{2A}}\). This will result in \(x = \frac{{-5 \pm \sqrt{{5^{2} - 4(1)(2)}}}}{{2(1)}}\).

Step 3: Evaluate under the square root

Evaluate under the square root in the formula: \(x = \frac{{-5 \pm \sqrt{{25 - 8}}}}{{2}} = \frac{{-5 \pm \sqrt{17}}}{2}\).

Step 4: Simplify and solve

Simplify the equation to get the final solutions \(x = \frac{{-5 + \sqrt{17}}}{2}\) and \(x = \frac{{-5 - \sqrt{17}}}{2}\), which are the solutions of the given quadratic equation.

Key Concepts

Solving Quadratic Equations

Grasping the technique to solve quadratic equations is pivotal for your mathematical journey. A quadratic equation is any equation in the form of \( ax^2 + bx + c = 0 \). Here, \(x\) represents an unknown variable, while \( a \), \( b \), and \( c \) are coefficients — with \( a \) not equal to zero. Why not zero? Because if \( a \) were zero, it would no longer be a quadratic equation but rather a linear one.

To solve these, we often use the quadratic formula, which is \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \). This formula is so clever that it considers all possible outcomes. The \( \pm \) symbol indicates that we'll get two solutions, since a parabola (the graph of a quadratic equation) usually intersects the x-axis at two points.

Remember to always double-check your solutions by plugging them back into the original equation. It's a foolproof way to ensure your answers make the quadratic equation balance out to zero, thus confirming their correctness.

Simplifying Irrational Numbers

When you encounter a square root of a non-perfect square in your solutions, that's when you're dealing with an irrational number. These numbers can't be neatly expressed as a simple fraction and their decimal forms go on forever without repeating. Classic examples include \(\sqrt{2}\), \(\sqrt{3}\), and in our case, \(\sqrt{17}\).

While we can't simplify \(\sqrt{17}\) into a neat integer or fraction, it's important to express your final answer in the simplest form possible. This may involve rationalizing denominators or simplifying expressions involving square roots. In the context of our equation, we leave it as \(\sqrt{17}\) since it's the simplest form. This expression is part of our final solutions to the quadratic equation, and even though it might look messy, it's the most accurate way to describe the answer.

Identifying Coefficients

Getting to grips with identifying coefficients is crucial to your success in using the quadratic formula. In a standard quadratic equation \( ax^2 + bx + c = 0 \), the coefficients are the numerical parts that multiply the variable terms. The coefficient \( a \) is in front of \( x^2 \), \( b \) is in front of \( x \), and \( c \) is the constant term, with no variable attached.

For the equation \( x^2 + 5x + 2 = 0 \), \( a \) equals 1 (it's invisible since anything times 1 is itself), \( b \) is 5, and \( c \) is 2. Identifying these correctly is a big step toward the correct application of the quadratic formula. Misidentify these, and your formula will churn out the wrong answers. So, take note of each coefficient's value: they are the key players in solving your quadratic equations.