Problem 67 - Introductory Algebra for College Students

Question

Explain how to solve $$x^{2}+6 x+8=0$$ using the quadratic formula,

Step-by-Step Solution

Answer

The solutions to the equation \(x^{2}+6 x+8=0\) are \(x = -2\) and \(x = -4\).

Step 1: Identify the coefficients

From the equation \(x^{2}+6 x+8=0\), we can identify the coefficients as follows: \(a = 1\), \(b = 6\) and \(c = 8\).

Step 2: Substitute the coefficients into the quadratic formula

Substitute \(a = 1\), \(b = 6\) and \(c = 8\) into the quadratic formula, yielding \[x = \frac{-6 \pm \sqrt{(6)^{2} - 4*(1)*(8)}}{2*(1)}\].

Step 3: Simplify the equation

Simplify the equation to get \[x = \frac{-6 \pm \sqrt{36 - 32}}{2}\].

Step 4: Calculate the square root and simplify further

After calculating the square root, the equation simplifies to: \[x = \frac{-6 \pm 2}{2}\].

Step 5: Finally, calculate x

The equation thus yields two solutions: \(x = \frac{-6 + 2}{2} = -2\) and \(x = \frac{-6 - 2}{2} = -4\).

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of the second degree. This means it has the general form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(aeq0\).
Quadratic equations are essential in various fields like physics, engineering, and economics due to their application in calculating different curves and parabolic trajectories. Here, the equation \(x^2 + 6x + 8 = 0\) is a typical quadratic equation.

Coefficients

In the context of a quadratic equation, the coefficients are the numbers \(a\), \(b\), and \(c\) in the equation \(ax^2 + bx + c = 0\).
They determine the shape and position of the parabola that represents the quadratic equation.

• \(a\): The coefficient of \(x^2\). It indicates the direction and width of the parabola. Here, \(a = 1\).
• \(b\): The coefficient of \(x\). This affects the symmetry of the parabola. In our equation, \(b = 6\).
• \(c\): The constant term. It impacts the vertical position of the parabola. For this equation, \(c = 8\).

Recognizing these coefficients is the first step in solving quadratic equations using methods like the quadratic formula.

Solving Equations

Solving equations is a fundamental concept in mathematics, crucial for determining unknown values in algebra.
For quadratic equations, one of the most effective methods is using the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].
This formula is versatile and can solve any quadratic equation, but it requires identifying the correct coefficients.
By substituting the coefficients \(a = 1\), \(b = 6\), and \(c = 8\) into the formula, we simplify and calculate to find both solutions for \(x\).
The solutions \(-2\) and \(-4\) are then derived, demonstrating the equation's potential solutions.

Mathematics Education

Understanding how to solve quadratic equations is a vital component of mathematics education.
This skill lays the groundwork for more advanced mathematical topics and problem-solving techniques.
Learning different methods to solve quadratic equations, like factoring, completing the square, and using the quadratic formula, enhances problem-solving skills.
The quadratic formula, in particular, is powerful as it works universally for any quadratic equation, provided the calculations are correct.

• It encourages analytical thinking as students learn to identify patterns and strategies applicable to diverse mathematical problems.
• Practicing these methods improves numerical fluency and comprehension, essential skills in both academic and real-world contexts.

Engaging with quadratic equations fosters a deeper appreciation of mathematical concepts and their applications.