For each quadratic equation given below, follow the steps outlined:
Arrange the equation into standard form 
Find the Discriminant using the formula and Determine the Nature of the Roots based on the value of the discriminant:
Prove by Solving the equation using the quadratic formula:

x^2-4x=- 1

A. Find the Discriminant:
B. Nature of the roots:
C: Prove by solving using the quadratic formula:



Let's work through the given quadratic equation step-by-step:

### Given equation:
\[
x^2 - 4x = -1
\]

### Step 1: Arrange the equation into standard form

To convert this into standard quadratic form, \(ax^2 + bx + c = 0\), we move the -1 from the right side to the left:

\[
x^2 - 4x + 1 = 0
\]

This is the standard form of the quadratic equation.

### Step 2: Find the Discriminant

The formula for the discriminant is:
\[
D = b^2 - 4ac
\]

In this equation, \(a = 1\), \(b = -4\), and \(c = 1\). Substituting these values into the discriminant formula:

\[
D = (-4)^2 - 4(1)(1)
\]
\[
D = 16 - 4 = 12
\]

### Step 3: Determine the Nature of the Roots

The nature of the roots depends on the value of the discriminant:
- If \(D > 0\), there are **two real and distinct** roots.
- If \(D = 0\), there are **two real and equal** roots.
- If \(D < 0\), there are **two complex** roots.

Since \(D = 12 > 0\), the equation has **two real and distinct roots**.

### Step 4: Solve Using the Quadratic Formula

The quadratic formula is:
\[
x = \frac{-b \pm \sqrt{D}}{2a}
\]

Substituting \(a = 1\), \(b = -4\), and \(D = 12\) into the formula:

\[
x = \frac{-(-4) \pm \sqrt{12}}{2(1)}
\]
\[
x = \frac{4 \pm \sqrt{12}}{2}
\]

Since \(\sqrt{12} = 2\sqrt{3}\), we get:

\[
x = \frac{4 \pm 2\sqrt{3}}{2}
\]

Simplifying:

\[
x = 2 \pm \sqrt{3}
\]

Thus, the two real roots are:
\[
x_1 = 2 + \sqrt{3}, \quad x_2 = 2 - \sqrt{3}
\]

### Summary:
- **A. Discriminant**: \(D = 12\)
- **B. Nature of the Roots**: Two real and distinct roots
- **C. Solved roots**: \(x = 2 + \sqrt{3}\) and \(x = 2 - \sqrt{3}\)

---

Would you like more details on any of the steps, or do you have any further questions?

Here are 5 related questions to expand on this:
1. What happens if the discriminant is zero?
2. Can you solve a quadratic equation by factoring instead of using the quadratic formula?
3. What are the real-life applications of quadratic equations?
4. How do you graph a quadratic equation?
5. What is the vertex form of a quadratic equation?

**Tip**: The discriminant is a quick way to determine the number and type of solutions without fully solving the equation.

For each quadratic equation given below, follow the steps outlined: Arrange the equation into standard form. Find the Discriminant using the formula and Determine the Nature of the Roots based on the value of the discriminant. Prove by Solving the equation using the quadratic formula: x^2 - 4x = -1. A. Find the Discriminant. B. Nature of the roots. C: Prove by solving using the quadratic formula.

Solve the quadratic equation x^2 - 4x + 1 = 0 using the quadratic formula. This step-by-step guide covers finding the discriminant (D = 12), determining the nature of the roots, and solving for the roots (x = 2 ± √3). Suitable for Grades 9-10.

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