Problem 27 - Algebra for College Students

Question

Solve each equation by using the quadratic formula. $$2 t^{2}-6 t+5=0$$

Step-by-Step Solution

Answer

The solutions are \(t = \frac{3}{2} + \frac{i}{2}\) and \(t = \frac{3}{2} - \frac{i}{2}\).

Step 1: Identify coefficients

Identify the coefficients in the quadratic equation a, b, and c in the standard form of a quadratic equation, \(a t^2 + bt + c = 0\).For the given equation, \(2t^2 - 6t + 5 = 0\), the coefficients are: a = 2, b = -6, c = 5.

Step 2: Write down the quadratic formula

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

Step 3: Substitute coefficients into the quadratic formula

Substitute the identified coefficients a = 2, b = -6, and c = 5 into the quadratic formula: \(t = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(2)(5)}}{2(2)}\).

Step 4: Simplify under the square root

First compute:\((-6)^2 = 36\)and\(4ac = 4 \cdot 2 \cdot 5 = 40\).Then substitute these values back into our quadratic formula:\(t = \frac{6 \pm \sqrt{36 - 40}}{4}\)

Step 5: Simplify the expression further

Simplify inside of the square root:\(t = \frac{6 \pm \sqrt{-4}}{4}\).

Step 6: Address the imaginary number

Since the square root of a negative number introduces an imaginary unit:\(\sqrt{-4} = 2i\).Then substitute back: \(t = \frac{6 \pm 2i}{4}\).

Step 7: Simplify the expression

Finally simplify the fraction:\(t = \frac{6}{4} \pm \frac{2i}{4}\) which further simplifies to:\(t = \frac{3}{2} \pm \frac{i}{2}\).

Key Concepts

Coefficients

In a quadratic equation of the form \(a t^2 + bt + c = 0\), the terms \(a, b,\) and \(c\) are known as coefficients. They are crucial for solving the equation using the quadratic formula. Let's identify them in our given quadratic equation:

For the equation \(2t^2 - 6t + 5 = 0\):
- The coefficient 'a' is 2.
- The coefficient 'b' is -6.
- The coefficient 'c' is 5.

Recognizing these coefficients is the first step when plugging values into the quadratic formula.

Imaginary Numbers

Sometimes, when solving quadratic equations, you may encounter negative numbers under the square root. These introduce imaginary numbers. In our example:

- We simplify under the square root to get \(\frac{6 \pm \sqrt{-4}}{4}\).
- The square root of -4 is \(2i\), where 'i' stands for the imaginary unit \(\text{with} i = \sqrt{-1}\).

Imaginary numbers are useful in extending the realm of solutions beyond the real numbers. In our case, after substituting the imaginary unit back, we obtained: \(t = \frac{6 \pm 2i}{4}\). This gives two complex solutions: \(\frac{3}{2} + \frac{i}{2}\) and \(\frac{3}{2} - \frac{i}{2}\).

Simplifying Expressions

Simplifying expressions is a key part of solving quadratic equations. It often involves breaking down complex fractions and radicals into simpler forms.

In our solution, we had the expression \(\frac{6 \pm 2i}{4}\). To simplify, follow these steps:
- Split the fraction: \(\frac{6}{4} \pm \frac{2i}{4}\).
- Simplify each part: \(\frac{6}{4} = \frac{3}{2}\) and \(\frac{2i}{4} = \frac{i}{2}\).

So, the final simplified solutions are \(t = \frac{3}{2} + \frac{i}{2}\) and \(t = \frac{3}{2} - \frac{i}{2}\).

Simplifying not only makes the result more readable but also ensures it is in its most concise form. Always perform simplification as a final step to confirm the accuracy and neatness of your solutions.