https://file.aiquickdraw.com/mathbot/file/17298548677584unxwxha.jpg

The equation in the image is:

\[
(x - a)(x - b) = 1
\]

You are asked whether this quadratic equation has two distinct real roots for any real numbers \(a\) and \(b\).

### Solution:
To solve this, we first need to rewrite the equation into standard quadratic form by expanding it:

\[
(x - a)(x - b) = 1
\]

Expanding the left-hand side:

\[
(x - a)(x - b) = x^2 - (a + b)x + ab
\]

So the equation becomes:

\[
x^2 - (a + b)x + ab = 1
\]

Now, subtract 1 from both sides to make it equal to 0:

\[
x^2 - (a + b)x + (ab - 1) = 0
\]

This is now in standard quadratic form:

\[
Ax^2 + Bx + C = 0
\]

Where:
- \(A = 1\)
- \(B = -(a + b)\)
- \(C = ab - 1\)

To determine whether this equation has two distinct real roots, we use the discriminant, \(\Delta\), given by:

\[
\Delta = B^2 - 4AC
\]

Substitute the values of \(A\), \(B\), and \(C\):

\[
\Delta = (-(a + b))^2 - 4(1)(ab - 1)
\]

Simplifying:

\[
\Delta = (a + b)^2 - 4(ab - 1)
\]
\[
\Delta = (a + b)^2 - 4ab + 4
\]
\[
\Delta = (a + b)^2 - 4ab + 4 = a^2 + 2ab + b^2 - 4ab + 4
\]
\[
\Delta = a^2 - 2ab + b^2 + 4
\]
\[
\Delta = (a - b)^2 + 4
\]

Since \((a - b)^2 \geq 0\) for any real values of \(a\) and \(b\), and we are adding 4 to a non-negative number, \(\Delta\) will always be positive. This means the discriminant \(\Delta > 0\) for all real \(a\) and \(b\).

### Conclusion:
The quadratic equation will always have two distinct real roots because the discriminant is always positive.

Would you like any further details or explanations? Here are some related questions you may find helpful:

1. How does the discriminant determine the nature of the roots of a quadratic equation?
2. What happens if the discriminant is zero?
3. Can the quadratic formula be used to find the exact roots of this equation?
4. How can the discriminant be related to the geometry of a parabola?
5. What are some real-life applications of quadratic equations?

Tip: Always check the discriminant when analyzing the roots of quadratic equations; it’s a quick way to predict their nature.

For any real numbers a and b, does the quadratic equation (x - a)(x - b) = 1 have two distinct real roots? Explain your answer.

Explore whether the quadratic equation (x - a)(x - b) = 1 has two distinct real roots for any real values of a and b. This detailed explanation covers key algebraic concepts like the quadratic equation, the discriminant, and the discriminant theorem, suitable for students in Grades 9-11.

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