https://file.aiquickdraw.com/mathbot/file/1728755657809ahafv8tu.jpg

The problems in the image ask to solve quadratic equations using the quadratic formula. The quadratic formula is:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Where \( a \), \( b \), and \( c \) are the coefficients from the quadratic equation of the form \( ax^2 + bx + c = 0 \).

Let’s solve one example and then you can choose more problems to work on if you want!

### Example 29: \( x^2 - 4x + 1 = 0 \)
- Here, \( a = 1 \), \( b = -4 \), and \( c = 1 \).

Using the quadratic formula:
\[
x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(1)}}{2(1)}
\]
\[
x = \frac{4 \pm \sqrt{16 - 4}}{2}
\]
\[
x = \frac{4 \pm \sqrt{12}}{2}
\]
\[
x = \frac{4 \pm 2\sqrt{3}}{2}
\]
\[
x = 2 \pm \sqrt{3}
\]

So the solutions are:
\[
x = 2 + \sqrt{3} \quad \text{or} \quad x = 2 - \sqrt{3}
\]

Would you like to see more examples solved in detail or ask about any particular equation? 

Here are some related questions for you:
1. Can you find the discriminant for equation 31?
2. What are the roots of equation 30 using the quadratic formula?
3. How does the value of \(b^2 - 4ac\) affect the type of solutions you get?
4. What is the solution for equation 35?
5. How do you simplify square roots in the quadratic formula?

**Tip:** Always check the discriminant (\(b^2 - 4ac\)) before proceeding, as it tells you if the roots are real or complex.

Use the quadratic formula to solve equations such as x^2 - 4x + 1 = 0, x^2 - 2x - 1 = 0, and more.

Learn how to solve quadratic equations like x^2 - 4x + 1 = 0 and x^2 - 2x - 1 = 0 using the quadratic formula. Step-by-step solutions include identifying coefficients and applying the formula. Suitable for Grades 8-10.

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