Find the real​ solutions, if​ any, of the following equation. Use the quadratic formula.
49t^2-14t+1=0 Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your answer.
A.
The solution set is StartSet nothing EndSetenter your response here.
​(Simplify your answer. Type an exact​ answer, using radicals as needed. Use a comma to separate answers as needed. Use integers or fractions for any numbers in the​ expression.)
B.
The solution is not a real number.

To solve the quadratic equation \(49t^2 - 14t + 1 = 0\) using the quadratic formula, we follow these steps:

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

In the equation \(49t^2 - 14t + 1 = 0\), we identify the coefficients:
\[ a = 49 \]
\[ b = -14 \]
\[ c = 1 \]

Next, we calculate the discriminant (\(\Delta\)):
\[ \Delta = b^2 - 4ac \]
\[ \Delta = (-14)^2 - 4(49)(1) \]
\[ \Delta = 196 - 196 \]
\[ \Delta = 0 \]

Since the discriminant is zero, there is exactly one real solution (a repeated root).

We substitute \(a\), \(b\), and \(\Delta\) into the quadratic formula:
\[ t = \frac{-(-14) \pm \sqrt{0}}{2(49)} \]
\[ t = \frac{14 \pm 0}{98} \]
\[ t = \frac{14}{98} \]
\[ t = \frac{1}{7} \]

Thus, the solution set is:
\[ \left\{ \frac{1}{7} \right\} \]

Therefore, the correct choice is:
**A. The solution set is \( \left\{ \frac{1}{7} \right\} \)**

Would you like further details or have any questions?

Here are five related questions to expand your understanding:
1. What are the steps to derive the quadratic formula from a standard quadratic equation?
2. How does the discriminant determine the nature of the roots of a quadratic equation?
3. Can a quadratic equation have complex solutions? If so, how are they calculated?
4. How can you complete the square to solve a quadratic equation?
5. What is the relationship between the coefficients of a quadratic equation and the sum and product of its roots?

**Tip:** Always check the discriminant first when solving quadratic equations, as it immediately tells you the number and type of solutions (real or complex).

Learn how to find the real solutions of the quadratic equation 49t^2 - 14t + 1 = 0 using the quadratic formula. Understand the discriminant and its role in determining the nature of the roots. Suitable for high school students studying quadratic equations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation