x²-10x+11=0 try solve in quadratic formula

To solve the quadratic equation \( x^2 - 10x + 11 = 0 \) using the quadratic formula, we follow these steps:

### Quadratic Formula:
The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a \), \( b \), and \( c \) are coefficients from the quadratic equation \( ax^2 + bx + c = 0 \).

### Step 1: Identify coefficients
For the equation \( x^2 - 10x + 11 = 0 \), the coefficients are:
- \( a = 1 \)
- \( b = -10 \)
- \( c = 11 \)

### Step 2: Substitute into the quadratic formula
\[
x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(1)(11)}}{2(1)}
\]

### Step 3: Simplify the expression
1. Calculate the discriminant:
   \[
   \sqrt{(-10)^2 - 4(1)(11)} = \sqrt{100 - 44} = \sqrt{56}
   \]
2. Simplify \( \sqrt{56} \):
   \[
   \sqrt{56} = \sqrt{4 \times 14} = 2\sqrt{14}
   \]
3. Substitute back:
   \[
   x = \frac{10 \pm 2\sqrt{14}}{2}
   \]

### Step 4: Simplify further
\[
x = \frac{10}{2} \pm \frac{2\sqrt{14}}{2}
\]
\[
x = 5 \pm \sqrt{14}
\]

### Final Solutions:
The solutions are:
\[
x_1 = 5 + \sqrt{14}
\]
\[
x_2 = 5 - \sqrt{14}
\]

These are the exact solutions in simplified form.

Would you like more details or have any questions?

Here are some related questions:
1. What is the significance of the discriminant in the quadratic formula?
2. How can you determine the nature of the roots using the discriminant?
3. How do you simplify square roots like \(\sqrt{56}\)?
4. What is the difference between exact and approximate solutions in quadratic equations?
5. Can the quadratic formula be used for non-real solutions?

**Tip:** Always check the discriminant (\(b^2 - 4ac\)) before solving; it tells you whether the roots are real or complex.

Learn how to solve the quadratic equation x² - 10x + 11 = 0 step-by-step with the quadratic formula. This problem explores algebraic concepts such as the quadratic formula and its application to find exact solutions. Suitable for high school students studying algebra.

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