The quadratic equation provided is:

$-3 = x^2 - 4x + 1$

Step 1: Rewrite the equation

Rearranging the equation into standard quadratic form $ax^2 + bx + c = 0$: $x^2 - 4x + 1 + 3 = 0 \implies x^2 - 4x + 4 = 0$

Here:

• $a = 1$
• $b = -4$
• $c = 4$

Step 2: Calculate the discriminant

The discriminant formula is:

$\Delta = b^2 - 4ac$

Substitute $a = 1$, $b = -4$, and $c = 4$:

$\Delta = (-4)^2 - 4(1)(4) = 16 - 16 = 0$

Step 3: Determine the number of real solutions

• If $\Delta > 0$: Two distinct real solutions.
• If $\Delta = 0$: One real solution.
• If $\Delta < 0$: No real solutions.

Since $\Delta = 0$, the equation has one real solution.

Final Answer:

1

Would you like a detailed explanation of any part of the solution?

Related Questions:

1. What happens to the discriminant if the value of $c$ is increased?
2. How is the nature of the roots determined from the discriminant?
3. How does the vertex form of the quadratic relate to the discriminant?
4. Can there be a case where a quadratic equation has exactly one real solution but not a perfect square trinomial?
5. How would the graph of this equation look?

Tip:

The discriminant is a quick and powerful way to determine the number and nature of solutions for quadratic equations. Use it before solving the equation to save time.