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 $x = 2 - \sqrt{3}$

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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.