Solve x2 = 5x -7 using the quadratic formula. What can you conclude about the nature of the solution?
 
X2 = 5x – 7
 
Writing in standard form
 
X2 - 5x + 7 = 0
 
(1 = A) (-5 = B) (7 = C)
 

 
 

 

 
 
X =
 
The quadratic equation has no real solutions
Is this correct?

To solve the quadratic equation \( x^2 = 5x - 7 \) using the quadratic formula, follow these steps:

### Step-by-Step Solution:

1. **Write the equation in standard form**:
   \[
   x^2 - 5x + 7 = 0
   \]
   Here, \( A = 1 \), \( B = -5 \), and \( C = 7 \).

2. **Quadratic Formula**:
   \[
   x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}
   \]
   Substitute \( A = 1 \), \( B = -5 \), and \( C = 7 \) into the formula.

3. **Calculate the discriminant**:
   \[
   \Delta = B^2 - 4AC
   \]
   \[
   \Delta = (-5)^2 - 4(1)(7)
   \]
   \[
   \Delta = 25 - 28
   \]
   \[
   \Delta = -3
   \]

4. **Evaluate the quadratic formula**:
   Since the discriminant (\( \Delta \)) is negative (\( \Delta = -3 \)), the solutions will be complex numbers.

   \[
   x = \frac{-(-5) \pm \sqrt{-3}}{2(1)}
   \]
   \[
   x = \frac{5 \pm \sqrt{-3}}{2}
   \]
   \[
   x = \frac{5 \pm i\sqrt{3}}{2}
   \]
   Thus, the solutions are:
   \[
   x = \frac{5 + i\sqrt{3}}{2} \quad \text{and} \quad x = \frac{5 - i\sqrt{3}}{2}
   \]

### Conclusion about the Nature of the Solution:
Since the discriminant (\( \Delta \)) is negative, the quadratic equation has two complex conjugate solutions. There are no real solutions.

### Additional Questions:

1. What is the discriminant and how does it determine the nature of the solutions of a quadratic equation?
2. How can you verify the solutions of a quadratic equation?
3. What are the implications of having complex solutions in real-world problems?
4. How does the quadratic formula change if the coefficient \( A \) is not 1?
5. What are other methods to solve quadratic equations apart from the quadratic formula?

### Tip:
Always check the discriminant first to quickly determine the nature of the solutions (real and distinct, real and equal, or complex).

Solve x^2 = 5x - 7 using the quadratic formula. What can you conclude about the nature of the solution?

Explore the solution to the quadratic equation x^2 - 5x + 7 using the quadratic formula. Learn how to calculate the discriminant and conclude the nature of the solutions, with step-by-step guidance for students in Grades 9-12. This problem involves complex numbers due to a negative discriminant.

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