Problem 41 - Elementary Algebra

Question

For the following problems, solve the equations using the quadratic formula. $$ b^{2}+3 b=-2 $$

Step-by-Step Solution

Answer

Answer: The solutions to the quadratic equation are approximately \(s_1 \approx 3.16\) and \(s_2 \approx -3.16\).

Step 1: Identify the equation and set it equal to zero

The given equation to solve is: $$ s^2 - 10 = 0 $$

Step 2: Isolate the squared term

Add 10 to both sides of the equation in order to isolate the squared term (s^2): $$ s^2 = 10 $$

Step 3: Apply the extraction of roots method

Taking the square root of both sides of the equation: $$ \sqrt{s^2} = \pm \sqrt{10} $$ Since the square root of a squared term results in the absolute value of the term, we have: $$ s = \pm \sqrt{10} $$

Step 4: Report the solutions

The solutions for the equation are the positive and negative roots of 10: $$ s_1 = \sqrt{10} \approx 3.16 $$ and $$ s_2 = -\sqrt{10} \approx -3.16 $$ So, the solutions for the given quadratic equation are approximately \(s_1 \approx 3.16\) and \(s_2 \approx -3.16\).

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental concept in algebra that describe a polynomial equation of degree two. The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). These equations are characterized by the highest exponent being 2, indicating the presence of a squared term.

When solving quadratic equations, you focus on finding the values of \(x\) that make the equation true, known as the roots or solutions of the equation. Quadratic equations can have two, one, or no real solutions, depending on the discriminant value \(b^2 - 4ac\):

• Two distinct real solutions if the discriminant is positive.
• One real solution (or a repeated root) if the discriminant is zero.
• No real solutions if the discriminant is negative.

Understanding quadratic equations is crucial because they model a variety of real-world situations, from projectile motion to economic forecasting. They also serve as a key foundation for more advanced math topics.

Solving Equations

Solving equations is a central skill in mathematics that involves finding the values for variables that satisfy the equation. For quadratic equations, there are several methods you can use:

• Factoring: Rewriting the equation in a product form \((x - p)(x - q) = 0\), so you can find solutions by setting each factor to zero.
• Completing the Square: Adjusting the equation into a perfect square trinomial, which can then be solved by taking the square root of both sides.
• Quadratic Formula: Utilizing the formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the solutions directly, which works for any quadratic equation.
• Extraction of Roots: Simplifying the equation to isolate the squared term and then taking the square root of both sides, as seen in the original problem used.

When solving equations, understanding different methods allows you to choose the most efficient one based on the equation's form. Practice is key to mastering these techniques and applying them effectively.

Square Roots

The concept of square roots is fundamental when dealing with quadratic equations and beyond. The square root of a number \(x\) is a value that, when multiplied by itself, gives \(x\). It is symbolized by \(\sqrt{x}\). For example, the square root of 9 is 3, because \(3 \times 3 = 9\).

Square roots can be positive or negative because squaring either a positive or negative number results in a positive number (e.g., \((3)^2 = 9\) and \((-3)^2 = 9\)). Therefore, when you extract the square root during equation solving, you must consider both the positive and negative solutions: \(\sqrt{x^2} = \pm x\).

Understanding square roots is critical in algebra, particularly in solving quadratic equations, as it directly relates to the extraction of roots. When applying the square root in these cases, it reveals important insights about the equation’s solutions, reflecting the symmetry around zero present in quadratic functions. Mastery of square roots opens up pathways not just in algebra, but in higher mathematics and various fields that apply mathematical principles.