Problem 32 - Algebra for College Students

Question

Solve each equation by using the quadratic formula. $$\frac{1}{4} x^{2}+\frac{17}{4}=2 x$$

Step-by-Step Solution

Answer

x = 4 + i or x = 4 - i.

Step 1: Bring the equation to standard form

The standard form of a quadratic equation is \[ax^2 + bx + c = 0\].First, move all terms to one side of the equation to set it to 0:\[\frac{1}{4} x^2 + \frac{17}{4} - 2x = 0.\]

Step 2: Combine like terms

Rewrite the equation as:\[\frac{1}{4} x^2 - 2x + \frac{17}{4} = 0.\]

Step 3: Identify coefficients

Identify the coefficients from the equation:\[a = \frac{1}{4}, \quad b = -2, \quad c = \frac{17}{4}.\]

Step 4: Write the quadratic formula

The quadratic formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.\]

Step 5: Substitute coefficients into the formula

Substitute the values of \(a\), \(b\), and \(c\) into the quadratic formula:\[x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot \frac{1}{4} \cdot \frac{17}{4}}}{2 \cdot \frac{1}{4}}.\]

Step 6: Simplify inside the square root

Simplify the discriminant part:\[(-2)^2 = 4, \quad 4 \cdot \frac{1}{4} \cdot \frac{17}{4} = \frac{17}{4}.\]So,\[b^2 - 4ac = 4 - \frac{17}{4} = \frac{16}{4} - \frac{17}{4} = -\frac{1}{4}.\]

Step 7: Compute the square root

Find the square root of the discriminant:\[\sqrt{-\frac{1}{4}} = \frac{i}{2}.\]

Step 8: Substitute back into the quadratic formula

Now, substitute back into the quadratic formula and solve:\[x = \frac{2 \pm \frac{i}{2}}{\frac{1}{2}}.\]Multiply through by 2 to clear the fraction:\[x = 4 \pm i.\]

Key Concepts

Solving Quadratic Equations

Solving quadratic equations might seem intimidating at first, but it's a valuable skill. The key tool for this is the quadratic formula. The general form of a quadratic equation is \(ax^2 + bx + c = 0\). The quadratic formula for finding the roots of this equation is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
First, you need to identify the coefficients \(a, b,\) and \(c\) from your equation.
For instance, in the equation \(\frac{1}{4} x^2 + \frac{17}{4} - 2x = 0\), the coefficients are \(a = \frac{1}{4}, b = -2,\) and \(c = \frac{17}{4}\).
Next, substitute these values into the quadratic formula, carefully following the arithmetic steps. This structured method allows you to find the solutions, whether they are real or complex numbers.
Practice regularly to become comfortable with these steps and using the quadratic formula. It's important to understand each step deeply to ensure you can handle any quadratic equation you encounter.

The Discriminant

The discriminant is a crucial part of the quadratic formula. It is found inside the square root and is given by \(b^2 - 4ac\).
The value of the discriminant determines the nature of the roots of the quadratic equation:

• If \(b^2 - 4ac \) is positive, there are two distinct real solutions.
• If \(b^2 - 4ac \) is zero, there is exactly one real solution (a repeated root).
• If \(b^2 - 4ac \) is negative, the solutions are complex or imaginary.

In our example, \(-2^2 - 4 \cdot \frac{1}{4} \cdot \frac{17}{4}\) gives a negative discriminant value of \(-\frac{1}{4}\). This indicates the presence of complex solutions. Calculating the discriminant is a vital step as it tells us what kind of solutions to expect, making it a key concept in solving quadratic equations.

Complex Solutions

Complex solutions arise when the discriminant of a quadratic equation is negative. In this case, as we saw, the discriminant was \(-\frac{1}{4}\). This means we have solutions involving the imaginary unit \(i\), where \(i = \sqrt{-1}\).
To handle complex solutions, we need to understand how to work with \(i\). For our example, with the discriminant as \(-\frac{1}{4}\), the square root becomes \ \frac{i}{2} \.
Substituting back into the quadratic formula, we get \(x = \frac{2 \pm \frac{i}{2}}{\frac{1}{2}}\), and simplifying gives \ x = 4 \pm i \.
Complex solutions always appear in conjugate pairs (e.g., \(4 + i\) and \(4 - i\)). Grasping this concept ensures you can tackle any quadratic equation, regardless of whether its solutions are real or complex. Practice identifying and solving for complex solutions will enhance your understanding and skill set significantly.