Problem 61 - Algebra 1

Question

Solve the quadratic equation. $$x^{2}-4 x=\frac{5}{6}$$

Step-by-Step Solution

Answer

The solutions of the quadratic equation are \(x = 4 + \sqrt{ \frac{68}{3}}/2\) and \(x = 4 - \sqrt{ \frac{68}{3}}/2\)

Step 1: Rearrange into Standard Form

Rearrange the quadratic equation into standard form, ax^2 + bx + c = 0. For the given problem, the equation becomes \(x^{2}-4 x - \frac{5}{6}= 0\)

Step 2: Identify a, b, and c

Identify a, b, and c from the rearranged equation. a is the coefficient of \(x^{2}\), b is the coefficient of x , and c is the constant. In our case, a=1, b=-4 and c=-\(\frac{5}{6}\)

Step 3: Apply Quadratic Formula

Applying the quadratic formula now, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} --> x = \frac{4 \pm \sqrt{(-4)^2 - 4*1*(-\frac{5}{6})}}{2*1}= \frac{4 \pm \sqrt{16+ \frac{20}{3}}}{2} = \frac{4 \pm \sqrt{ \frac{68}{3}}}{2}\)

Step 4: Solve

Finally solve the expression obtained to get the solution for x, which are \(x = 4 + \sqrt{ \frac{68}{3}}/2\) and \(x = 4 - \sqrt{ \frac{68}{3}}/2\)

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions of a quadratic equation, which is any equation that can be written in the form \( ax^2 + bx + c = 0 \). The quadratic formula is represented as: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] This formula allows us to find the values of \( x \) that make the equation true. The expression under the square root sign, \( b^2 - 4ac \), is known as the discriminant. The discriminant can tell us about the nature of the roots:

• If \( b^2 - 4ac > 0 \), there are two distinct real roots.
• If \( b^2 - 4ac = 0 \), there is one real root (a repeated root).
• If \( b^2 - 4ac < 0 \), the equation has no real roots (the roots are complex).

The quadratic formula is essential for solving any quadratic equation, whether its roots are real or complex.

Standard Form of Quadratic Equation

The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \). It is important for the equation to be in this form when using the quadratic formula. Here, \( a \), \( b \), and \( c \) are coefficients, where \( a eq 0 \). The coefficient \( a \) is associated with the term \( x^2 \), \( b \) with \( x \), and \( c \) is the constant term. To convert any quadratic equation into its standard form, all terms should be on one side of the equation, set equal to zero.
For example, consider the equation originally given: \( x^2 - 4x = \frac{5}{6} \) To rearrange it to standard form, move \( \frac{5}{6} \) to the left: \( x^2 - 4x - \frac{5}{6} = 0 \) This step ensures that the equation is ready to be solved using the quadratic formula or other techniques.

Solving Quadratic Equations Step by Step

Solving quadratic equations step-by-step involves identifying and applying the appropriate methods based on the form of the equation. Here is a typical approach:1. **Rearrange the Equation (Standard Form)**: Ensure the equation is in the standard form \( ax^2 + bx + c = 0 \). This involves moving all terms to one side of the equation, as demonstrated in the provided example. 2. **Identify Coefficients**: Clearly identify the coefficients \( a \), \( b \), and \( c \). These values are essential for using the quadratic formula. In our example, \( a = 1 \), \( b = -4 \), and \( c = -\frac{5}{6} \). 3. **Apply the Quadratic Formula**: Substitute the coefficients into the quadratic formula. Given the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), plug in the identified values. This yields the expression: \( x = \frac{4 \pm \sqrt{(-4)^2 - 4 \times 1 \times -\frac{5}{6}}}{2 \times 1} \). 4. **Solve the Expression**: Carefully solve the expression to find the values of \( x \), which are the solutions to the quadratic equation. Follow each step to arrive at the simplest form of the solution. In this example, the solutions are determined as \( x = \frac{4 \pm \sqrt{\frac{68}{3}}}{2} \). By following these steps, you can confidently solve any quadratic equation, reinforcing your mathematical skills and understanding.