To solve the problem of factoring the expression [tex]\( x^2 - 8xy + 15y^2 \)[/tex], follow these steps:
1. Identify and understand the form: The expression [tex]\( x^2 - 8xy + 15y^2 \)[/tex] is a quadratic in terms of [tex]\( x \)[/tex] with [tex]\( y \)[/tex] also involved.
2. Looking for the factorization: We seek to factor this expression into a product of two binomials:
[tex]\[(x + Ay)(x + By),\][/tex]
where [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are constants we wish to determine.
3. Expand the factorized form to find coefficients:
[tex]\[(x + Ay)(x + By) = x^2 + (A + B)xy + AB \cdot y^2.\][/tex]
4. Compare coefficients with the original expression:
So, we need
[tex]\[
A + B = -8,\quad AB = 15.
\][/tex]
5. Solve the system of equations:
[tex]\[
A + B = -8 \quad \text{and} \quad AB = 15.
\][/tex]
By solving, we can find values of [tex]\( A \)[/tex] and [tex]\( B \)[/tex] that satisfy both conditions. We observe that the pairs of factors of 15 are:
[tex]\[
(1, 15),\quad (-1, -15), \quad (3, 5), \quad (-3, -5).
\][/tex]
By testing these pairs, we find that:
[tex]\[
-3 + (-5) = -8 \quad \text{and} \quad (-3) \cdot (-5) = 15.
\][/tex]
Therefore, the correct values are [tex]\( -3 \)[/tex] and [tex]\( -5 \)[/tex].
6. Write the factored form:
[tex]\[
x^2 - 8xy + 15y^2 = (x - 3y)(x - 5y).
\][/tex]
7. Verify with given options:
Comparing with the options provided:
- A [tex]\( (x - 15y)(x - y) \)[/tex]
- B [tex]\( (x + 15y)(x + y) \)[/tex]
- C [tex]\( (x + 5y)(x + 3y) \)[/tex]
- D [tex]\( (x - 5y)(x - 3y) \)[/tex]
The correct factorization matches option D [tex]\((x - 5y)(x - 3y)\)[/tex].
Therefore, the correct answer is:
D [tex]\( (x - 5y)(x - 3y) \)[/tex].
Click the "Next" button to proceed.
