Problem 28 - Algebra for College Students

Question

Solve each equation by using the quadratic formula. $$2 y^{2}+1=2 y$$

Step-by-Step Solution

Answer

The solutions are \( y = \frac{1}{2} \pm \frac{i}{2} \).

Step 1: Rewrite the Equation in Standard Form

Start by rewriting the given equation in the standard form of a quadratic equation, which is a y^2 + b y + c = 0.The given equation is 2y^2 + 1 = 2y.Subtract 2y from both sides to get it into the standard form:2y^2 - 2y + 1 = 0.

Step 2: Identify Coefficients

Identify the coefficients a, b, and c from the quadratic equation 2 y^2 - 2 y + 1 = 0:a = 2,b = -2,c = 1.

Step 3: Set Up the Quadratic Formula

Use the quadratic formula to solve the equation. The quadratic formula is:\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Plug in the coefficients a, b, and c:\[ y = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(2)(1)}}{2(2)} \]

Step 4: Compute the Discriminant

Calculate the discriminant b^2 - 4ac:(-2)^2 - 4(2)(1) = 4 - 8 = -4.Since the discriminant is negative, the solutions will be complex numbers.

Step 5: Solve for y

Finish solving the quadratic formula:\[ y = \frac{2 \pm \sqrt{-4}}{4} \]Simplify the square root of the negative number using imaginary unit i where \( \sqrt{-4} = 2i \):\[ y = \frac{2 \pm 2i}{4} \rightarrow y = \frac{2}{4} \pm \frac{2i}{4} \rightarrow y = \frac{1}{2} \pm \frac{i}{2} \]

Key Concepts

Standard Form

The standard form of a quadratic equation is important for solving equations using the quadratic formula. In standard form, a quadratic equation looks like this:
a y^2 + b y + c = 0
Here, a, b, and c are constants.
To rewrite any quadratic equation in this form, we need to move all terms to one side of the equation, setting the other side to zero.
In our example, we had the equation:2 y^2 + 1 = 2 y. By subtracting 2 y from both sides, we obtain:2 y^2 - 2 y + 1 = 0.
Getting the equation in this form helps us identify the coefficients we need for the quadratic formula.

Complex Numbers

Complex numbers are used when we have solutions to equations that cannot be represented by real numbers alone.
A complex number has two parts: a real part and an imaginary part, written in the form a + bi, where: a is the real part, and bi is the imaginary part (with i being the imaginary unit).
In the example, the discriminant was negative, which indicated we would get complex solutions. Our final solutions y = 1/2 ± i/2 are an example of complex numbers.
These kinds of solutions show up in many fields, and knowing how to work with them is a valuable skill.

Discriminant

The discriminant is a key part of the quadratic formula and tells us about the nature of the roots of the quadratic equation.
It's found inside the square root of the quadratic formula: b^2 - 4ac.
The value of the discriminant helps us determine the type of solutions we can expect:

• If the discriminant is positive, we get two distinct real roots.


• If the discriminant is zero, we get exactly one real root (a repeated root).


• If the discriminant is negative, like in our example (-4), we get two complex roots.

Knowing this helps us expect what kind of solutions we will find even before solving the equation.

Imaginary Unit

When the discriminant is negative, the solutions to the quadratic equation involve the imaginary unit, denoted by i.
The imaginary unit is defined as i = sqrt{-1). It allows us to take square roots of negative numbers and represents an essential expansion of the number system.
In our example, we encountered a square root of -4. Tackling this involves using i: sqrt{-4) = 2i.
Using the imaginary unit allows us to simplify expressions involving negative square roots and is necessary for understanding complex solutions.
The imaginary unit helps us explore solutions to equations that real numbers alone can't address. It opens up new areas in mathematics, physics, and engineering.