Problem 36 - Basic Technical Mathematics with Calculus

Question

Solve the given quadratic equations, using the quadratic formula. Exercises \(5-8\) are the same as Exercises \(11-14\) of Section 7.2. $$c^{2} x^{2}-x-1=x^{2}$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = \frac{1 \pm \sqrt{4c^2 - 3}}{2(c^2 - 1)}\).

Step 1: Move all terms to one side of the equation

Start by moving all terms to the left side of the equation to set it equal to 0. Subtract \(x^2\) from both sides:\[c^2 x^2 - x - 1 - x^2 = 0\]

Step 2: Simplify the equation

Combine like terms in the equation. Notice that \(c^2 x^2\) and \(-x^2\) can be combined:\[(c^2 - 1)x^2 - x - 1 = 0\]

Step 3: Identify coefficients for the quadratic formula

In the quadratic equation \(ax^2 + bx + c = 0\), the coefficients are identified as follows:- \(a = c^2 - 1\)- \(b = -1\)- \(c = -1\)

Step 4: Apply the quadratic formula

The quadratic formula is \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. Substitute the values for \(a\), \(b\), and \(c\) into the formula:\[x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(c^2 - 1)(-1)}}{2(c^2 - 1)}\]Simplify:\[x = \frac{1 \pm \sqrt{1 + 4(c^2 - 1)}}{2(c^2 - 1)}\]

Step 5: Simplify the expression under the square root

Simplify the expression inside the square root:\[1 + 4(c^2 - 1) = 1 + 4c^2 - 4\]Further simplify:\[4c^2 - 3\]

Step 6: Write the final solution

Substitute back the simplified expression under the square root into the quadratic formula's solution:\[x = \frac{1 \pm \sqrt{4c^2 - 3}}{2(c^2 - 1)}\]

Key Concepts

Understanding Quadratic Equations

A quadratic equation is a type of polynomial equation that involves the variable raised to the power of two, represented as \(ax^2 + bx + c = 0\). These equations are called 'quadratic' because 'quad' stands for 'square.' The equation typically contains three terms:

• The squared term, \(ax^2\), which determines the parabola's opening direction and width.
• The linear term, \(bx\), which affects the position of the axis of symmetry.
• A constant, \(c\), which dictates the parabola's position relative to the y-axis.

Quadratic equations are integral in algebra and a vital part of many real-life applications, ranging from physics problems to optimizing business models. In our solved exercise, you can see how understanding the equation's format is crucial for tackling problems using methods like the quadratic formula.

Solving Quadratic Equations Using the Quadratic Formula

To effectively solve quadratic equations, especially those that cannot be easily factored, you can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]It is a universal method that works for any quadratic equation, providing solutions whether the roots are real or complex.
To apply it, follow these simple steps:

• First, ensure the equation is in the standard form \(ax^2 + bx + c = 0\).
• Next, identify the coefficients \(a\), \(b\), and \(c\).
• Finally, substitute these values into the quadratic formula.

The formula helps find the roots, or solutions, of the equation, which are the points where the graph of the quadratic function intersects the x-axis. In our problem, by correctly identifying and substituting the coefficients, we can solve for \(x\) effectively using this method.

Mathematics Education and Quadratic Equations

Quadratic equations form a foundational part of mathematics education, appearing in curriculums worldwide. They teach students critical thinking and problem-solving skills, setting the foundation for higher-level math and science.
Understanding how to rearrange complex problems and apply formulae like the quadratic formula helps develop strategic methods to find solutions. In educational settings, exercises like the one we solved help students to:

• Strengthen algebraic manipulation skills, essential for tackling more complex equations in advanced topics.
• Develop an understanding of functions and their graphical representations, which is key in calculus and other mathematical fields.
• Build confidence in handling abstract mathematical concepts.

Consistent practice with quadratic equations enhances a student's analytical abilities, preparing them for various academic and real-world challenges.

Exploring Algebra Through Quadratic Equations

Algebra is the mathematical language that deals with symbols and the rules for manipulating these symbols. It lays the groundwork for understanding and designing complex mathematical models and theories.
Quadratic equations, as part of algebra, involve methods that go beyond mere arithmetic calculations. Key components of learning algebra through quadratic equations include:

• Recognizing patterns and establishing relationships among variables.
• Understanding that equations represent a balance, where both sides are equal, guiding the transformation of equations.
• Learning methods such as completing the square, factoring, and using the quadratic formula, offering multiple tools to approach problems.

By exploring algebra through quadratic equations, students can view mathematical problems from various perspectives, using logical reasoning to derive solutions while appreciating the elegance and utility of mathematics.