Problem 3 - Introductory Algebra for College Students

Question

Solve each equation using the quadratic formula. Simplify irrational solutions, if possible. $$x^{2}+5 x+3=0$$

Step-by-Step Solution

Answer

The solutions of the equation \(x^{2}+5 x+3=0 \) are \( x = -3.302775637732\) and \( x = -0.697224362268\)

Step 1: Identify a, b, c

The equation is \(x^{2}+5 x+3=0 \). A quadratic equation is in the form \(ax^{2}+bx+c=0\), comparing with which we get \( a=1, b=5, c=3 \).

Step 2: Calculate the Discriminant

The Discriminant is given by \(b^{2}-4ac\). Plugging in the values \(a=1, b=5, c=3 \), calculate \(D = b^{2}-4ac = 5^{2}-4(1)(3) = 25 - 12 = 13\).

Step 3: Apply the Quadratic Formula

The quadratic formula is given by \(x = [-b ± sqrt(D)] / (2a)\), where D is the Discriminant calculated in Step 2. Plugging in the values, we get \(x = [-5 ± sqrt(13)] / 2\). This gives two solutions \( x1 = [-5 - sqrt(13)] / 2 \) and \( x2 = [-5 + sqrt(13)] / 2 \).

Step 4: Simplify

Simplify \( x1 \) and \( x2 \) to get the final solutions. Simplifying we get the roots \( x1 = -3.302775637732\) and \( x2 = -0.697224362268\).

Key Concepts

Discriminant

In a quadratic equation of the form \(ax^{2} + bx + c = 0\), the discriminant is a key component. It is represented by \(D = b^{2} - 4ac\), and it plays a crucial role in determining the nature of the equation's solutions. Think of the discriminant as a numerical tool that tells you about the roots of the equation without actually solving it.

• If \(D > 0\), the quadratic equation will have two distinct real solutions.
• If \(D = 0\), it will have one real solution (or two identical real roots).
• If \(D < 0\), the equation will have no real solutions, only complex ones.

In the original exercise, we calculated the discriminant \(D\) to be \(13\). Since \(13 > 0\), this tells us the equation \(x^{2} + 5x + 3 = 0\) has two distinct real solutions.

Irrational Solutions

Irrational solutions are solutions that cannot be expressed as a simple fraction or a decimal that terminates or repeats. They often involve square roots of non-perfect squares, resulting in non-repeating and non-terminating decimals.

In the quadratic formula \(x = \frac{-b \pm \sqrt{D}}{2a}\), when the discriminant \(D\) is not a perfect square, the solutions will be irrational. For example, in the solution process of \(x^{2} + 5x + 3 = 0\), the discriminant \(13\) is not a perfect square. Thus, the resulting solutions \(x_1 = \frac{-5 - \sqrt{13}}{2}\) and \(x_2 = \frac{-5 + \sqrt{13}}{2}\) are irrational.

It's important to simplify irrational solutions as much as possible for clarity and precision, ensuring solutions such as these showcase their exact form rather than an approximate decimal value.

Quadratic Equation

The quadratic equation is a standard form of a polynomial equation of degree two, typically given by \(ax^{2} + bx + c = 0\). The solutions to this equation are the values of \(x\) that make the equation true, and they can be found using various methods such as factoring, completing the square, or most effectively, the quadratic formula.

The quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) offers a systematic way to find these solutions, encompassing both real and complex ones. This formula provides the root values based on the coefficients \(a\), \(b\), \(c\), and the computed discriminant \(b^{2} - 4ac\).

Quadratic equations arise in various real-world scenarios, including physics problems, engineering challenges, and financial calculations. Understanding their properties and solution methods like the quadratic formula allows for easy determination of critical values such as maximum or minimum points, intersection points, and optimal solutions.