Problem 74 - Elementary Algebra

Question

Problem: Use the quadratic formula to solve \(2 x^{2}-7 x-1=0 .\) Incorrect Answer: \(x=\frac{-(-7) \pm \sqrt{(-7)^{2}-4(2)(-1)}}{2(2)}\) $$ \begin{aligned} &x=\frac{7 \pm \sqrt{49-8}}{4} \\ &x=\frac{7 \pm \sqrt{41}}{4} \end{aligned} $$ 75\. Problem: Use the quadratic formula to solve \(x^{2}-30=0\). Incorrect Answer: \(x=\frac{-(-30) \pm \sqrt{(-30)^{2}-4(1)(0)}}{2(1)}\) \(x=\frac{30 \pm \sqrt{900-0}}{2}\) \(x=\frac{30 \pm \sqrt{900}}{2}\) \(x=\frac{30 \pm 30}{2}\) \(x=\frac{30+30}{2}\) or \(x=\frac{30-30}{2}\) \(x=\frac{60}{2} \quad\) or \(\quad x=\frac{0}{2}\) \(x=30 \quad\) or \(\quad x=0\)

Step-by-Step Solution

Answer

The solutions are \( x = \frac{7 + \sqrt{57}}{4} \) and \( x = \frac{7 - \sqrt{57}}{4} \).

Step 1 - Identify the coefficients

For the equation \( 2x^{2} - 7x - 1 = 0 \), identify the coefficients: \( a = 2 \), \( b = -7 \), \( c = -1 \).

Step 2 - Write down the quadratic formula

The quadratic formula is \( x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \).

Step 3 - Substitute the coefficients into the formula

Substitute \( a = 2 \), \( b = -7 \), and \( c = -1 \) into the formula: \( x = \frac{-(-7) \pm \sqrt{(-7)^{2} - 4(2)(-1)}}{2(2)} \).

Step 4 - Simplify inside the square root

Simplify inside the square root: \( (-7)^{2} = 49 \) and \( 4(2)(-1) = -8 \). This makes the formula: \( x = \frac{7 \pm \sqrt{49 + 8}}{4} \).

Step 5 - Simplify inside the square root further

Continue simplifying inside the square root: \( 49 + 8 = 57 \), so the equation is \( x = \frac{7 \pm \sqrt{57}}{4} \).

Step 6 - Write the final solutions

The solutions to the equation are \( x = \frac{7 + \sqrt{57}}{4} \) and \( x = \frac{7 - \sqrt{57}}{4} \).

Key Concepts

quadratic equations

Quadratic equations are a type of polynomial equation that have the general form of \(ax^2 + bx + c = 0\). They are called 'quadratic' because the highest power of the variable \(x\) is 2.
The coefficients \(a\), \(b\), and \(c\) are real numbers, and \(a eq 0\) because if \(a\) were 0, the equation would no longer be quadratic but linear.
Examples of quadratic equations include:

• \(x^2 - 4x + 3 = 0\)
• \(2x^2 + 3x - 2 = 0\)
• \(5x^2 + x - 7 = 0\)

Quadratic equations can appear in various real-world contexts, such as in physics for projectile motion, in business for calculating profit or loss, and in many other areas.
To solve these equations, we need methods that can handle the square term, which is quadratic in nature.
One popular and effective method is using the quadratic formula which provides a straightforward way to find the solutions, or roots, of the equation.

solving equations

Solving equations, especially quadratic ones, means finding the values of the variable that make the equation true. For a quadratic equation, there are several standard methods to find these solutions:

• **Factoring**: This method involves expressing the quadratic equation in terms of its factors. For instance, \(x^2 - 5x + 6 = 0\) can be factored as \((x - 2)(x - 3) = 0\).
• **Completing the square**: This means rewriting the equation in the form \((x - p)^2 = q\) and then solving for \(x\).
• **Quadratic formula**: This is the most general method and works for all types of quadratic equations. The quadratic formula is:
  \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

Using the quadratic formula involves identifying the coefficients \(a\), \(b\), and \(c\), substituting them into the formula, and then simplifying to find the values of \(x\). Let's break it down with an example:
For the equation \(2x^2 - 7x - 1 = 0\), the steps you'd take are:

• Identify \(a = 2\), \(b = -7\), and \(c = -1\).
• Substitute into the formula: \(x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(2)(-1)}}{2(2)}\).
• Simplify inside the square root: \(\sqrt{49 + 8} = \sqrt{57}\).
• Find the final solutions: \(x = \frac{7 \pm \sqrt{57}}{4}\).

This provides two solutions for \(x\): \(\frac{7 + \sqrt{57}}{4}\) and \(\frac{7 - \sqrt{57}}{4}\). Each step is crucial to arriving at the correct answers.

algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. In elementary algebra, these symbols (often represented as letters) represent numbers in equations and formulas. Key components of algebra include:

• **Variables**: Symbols (like \(x\) or \(y\)) that stand for unknown values.
• **Constants**: Numbers that have fixed values.
• **Coefficients**: Numbers that multiply variables, such as the \(2\) in \(2x\).
• **Exponents**: Indicate repeated multiplication, such as the \(2\) in \(x^2\).

Algebra helps us model real-world situations through equations and expressions. By manipulating these equations, we can solve for unknown values:

• **Equations**: Statements that show the equality of two expressions, such as \(2x^2 - 7x - 1 = 0\).
• **Expressions**: Combinations of numbers, variables, and operators, like \(3x + 2\).

Algebra is foundational for advanced topics in mathematics and science. Mastery of algebraic manipulation is essential for solving all kinds of mathematical problems, from simple arithmetic to complex calculus.
Understanding quadratic equations and their solutions is an important part of algebra. It provides the tools needed to solve more complex problems. With practice, solving quadratic equations and transforming algebraic expressions becomes much easier.