Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. $$ \text { Solve: } 2 x^{2}-7 x-1=0 $$

Short answer

\(x = \frac{7 \pm \sqrt{57}}{4}\)

Step-by-step solution

Identify the quadratic equation

The given equation is in the standard quadratic form: \[ 2x^2 - 7x - 1 = 0 \] where, \(a = 2\), \(b = -7\), and \(c = -1\).

Use the quadratic formula

The quadratic formula is used to solve for \(x\): \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Substitute \(a = 2\), \(b = -7\), and \(c = -1\) into the formula.

Calculate the discriminant

First, find the discriminant, \(b^2 - 4ac\): \[ (-7)^2 - 4(2)(-1) = 49 + 8 = 57 \]

Substitute the discriminant back into the quadratic formula

Substitute \(\sqrt{57}\) back into the quadratic formula: \[ x = \frac{-(-7) \pm \sqrt{57}}{2(2)} \] which simplifies to: \[ x = \frac{7 \pm \sqrt{57}}{4} \]

Write the final solutions

The solutions to the quadratic equation are: \[ x = \frac{7 + \sqrt{57}}{4} \] and \[ x = \frac{7 - \sqrt{57}}{4} \]

Key concepts

quadratic formula

The quadratic formula is a powerful tool for solving quadratic equations. This formula provides a straightforward method to find the roots of any quadratic equation, even if it cannot be factored easily.
The general form of a quadratic equation is given by: \[ ax^2 + bx + c = 0 \]
This equation has coefficients:

• \(a\) is the coefficient of \(x^2\)
  
• \(b\) is the coefficient of \(x\)
  
• \(c\) is the constant term
  

The quadratic formula to find the roots is:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula can be used when you know the values of \(a\), \(b\), and \(c\). Simply substitute these values into the formula to calculate the roots.

discriminant

The discriminant is a part of the quadratic formula that indicates the nature of the roots. It is found inside the square root symbol and is given by:
\[ b^2 - 4ac \]
The value of the discriminant helps determine the number and type of roots:

• If the discriminant is greater than 0, the quadratic equation has two distinct real roots.
  
• If the discriminant equals 0, the quadratic equation has exactly one real root (a repeated root).
  
• If the discriminant is less than 0, the quadratic equation has two complex roots.
  

In the example provided, the discriminant is calculated as follows:
\[ (-7)^2 - 4(2)(-1) = 49 + 8 = 57 \]
Since 57 is greater than 0, the quadratic equation has two distinct real roots.

solving quadratic equations

Solving quadratic equations involves finding the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). Here’s a step-by-step guide on how to use the quadratic formula to solve such equations:
First, identify the coefficients \(a\), \(b\), and \(c\) from the quadratic equation.
Next, calculate the discriminant, \(b^2 - 4ac\).
Substitute the coefficients and the discriminant value into the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Simplify the expression to find the two potential values of \(x\), which represent the roots of the equation.
Using our example:

• Substitute \(a = 2\), \(b = -7\), and \(c = -1\) into the formula.
  
• Calculate the discriminant, which we already found to be 57.
  
• Substitute back into the formula:
  \[ x = \frac{7 \pm \sqrt{57}}{4} \]
  This leads us to the solutions:
  
• \[ x = \frac{7 + \sqrt{57}}{4} \] and
  \[ x = \frac{7 - \sqrt{57}}{4} \]
  

algebra

Algebra involves the study of mathematical symbols and rules for manipulating these symbols. It's the foundational branch of mathematics used to solve equations and represent abstract concepts.
Quadratic equations are a key topic in algebra. These equations often appear in various problems, from simple calculations to more complex scenarios.
Understanding quadratic equations and the quadratic formula is essential for solving many algebraic problems. With algebra, you can:

• Manipulate equations to isolate variables
  
• Understand relationships between variables
  
• Simplify complex expressions
  

Algebraic skills are not only useful for academic purposes but also have practical applications in fields like engineering, technology, and finance.

roots of equations

The roots of an equation are the values of the variable that make the equation true. For quadratic equations of the form \(ax^2 + bx + c = 0\), the roots can be found using the quadratic formula or factoring.
Let's focus on using the quadratic formula method:
From the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula provides two solutions, commonly referred to as the roots of the quadratic equation.
In our example
\[ 2x^2 - 7x - 1 = 0 \], the discriminant was 57, indicating two distinct real roots.
The roots are then:

• \[ x = \frac{7 + \sqrt{57}}{4} \]
  
• \[ x = \frac{7 - \sqrt{57}}{4} \]
  

Understanding the roots helps in analyzing the behavior of quadratic functions and their graphs.