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Answer :
We start with the expression
[tex]$$
625y^2 + 400y - 36 + 20z - z^2.
$$[/tex]
Step 1. Group the terms in [tex]\( y \)[/tex] and [tex]\( z \)[/tex].
Separate the expression into two groups:
[tex]$$
\underbrace{625y^2 + 400y - 36}_{\text{terms in } y} \quad \text{and} \quad \underbrace{20z - z^2}_{\text{terms in } z}.
$$[/tex]
Step 2. Rewrite the [tex]\( y \)[/tex]-terms as a completed square.
Notice that
[tex]$$
(25y + 8)^2 = 625y^2 + 400y + 64.
$$[/tex]
This is very similar to the [tex]\( y \)[/tex]-terms in our expression. To correct for the difference in the constant term, write:
[tex]$$
625y^2 + 400y - 36 = (25y + 8)^2 - 100,
$$[/tex]
since
[tex]$$
(25y + 8)^2 - 100 = \left(625y^2 + 400y + 64\right) - 100 = 625y^2 + 400y - 36.
$$[/tex]
Step 3. Rewrite the [tex]\( z \)[/tex]-terms as a completed square.
For the [tex]\( z \)[/tex]-terms, first factor out a negative sign:
[tex]$$
20z - z^2 = -\left(z^2 - 20z\right).
$$[/tex]
Now, complete the square for [tex]\( z^2 - 20z \)[/tex] by noticing that:
[tex]$$
(z - 10)^2 = z^2 - 20z + 100.
$$[/tex]
Thus, we can write:
[tex]$$
z^2 - 20z = (z - 10)^2 - 100,
$$[/tex]
and so
[tex]$$
20z - z^2 = -\left[(z - 10)^2 - 100\right] = 100 - (z - 10)^2.
$$[/tex]
Step 4. Combine the rewritten parts.
Substitute the rewritten forms into the original expression:
[tex]\[
\begin{aligned}
625y^2 + 400y - 36 + 20z - z^2 &= \Bigl[(25y + 8)^2 - 100\Bigr] + \Bigl[100 - (z-10)^2\Bigr] \\
&= (25y + 8)^2 - (z-10)^2.
\end{aligned}
\][/tex]
Now the expression is written as a difference of two squares.
Step 5. Factor the difference of squares.
Recall that a difference of two squares factors as:
[tex]$$
A^2 - B^2 = (A - B)(A + B).
$$[/tex]
Here, let:
[tex]$$
A = 25y + 8 \quad \text{and} \quad B = z - 10.
$$[/tex]
Thus, the expression factors as:
[tex]\[
\begin{aligned}
(25y + 8)^2 - (z-10)^2 &= \Bigl[(25y + 8) - (z-10)\Bigr]\, \Bigl[(25y + 8) + (z-10)\Bigr] \\
&= (25y + 8 - z + 10)(25y + 8 + z - 10) \\
&= (25y - z + 18)(25y + z - 2).
\end{aligned}
\][/tex]
Final Answer:
The factored form of the expression is
[tex]$$
\boxed{(25y - z + 18)(25y + z - 2)}.
$$[/tex]
[tex]$$
625y^2 + 400y - 36 + 20z - z^2.
$$[/tex]
Step 1. Group the terms in [tex]\( y \)[/tex] and [tex]\( z \)[/tex].
Separate the expression into two groups:
[tex]$$
\underbrace{625y^2 + 400y - 36}_{\text{terms in } y} \quad \text{and} \quad \underbrace{20z - z^2}_{\text{terms in } z}.
$$[/tex]
Step 2. Rewrite the [tex]\( y \)[/tex]-terms as a completed square.
Notice that
[tex]$$
(25y + 8)^2 = 625y^2 + 400y + 64.
$$[/tex]
This is very similar to the [tex]\( y \)[/tex]-terms in our expression. To correct for the difference in the constant term, write:
[tex]$$
625y^2 + 400y - 36 = (25y + 8)^2 - 100,
$$[/tex]
since
[tex]$$
(25y + 8)^2 - 100 = \left(625y^2 + 400y + 64\right) - 100 = 625y^2 + 400y - 36.
$$[/tex]
Step 3. Rewrite the [tex]\( z \)[/tex]-terms as a completed square.
For the [tex]\( z \)[/tex]-terms, first factor out a negative sign:
[tex]$$
20z - z^2 = -\left(z^2 - 20z\right).
$$[/tex]
Now, complete the square for [tex]\( z^2 - 20z \)[/tex] by noticing that:
[tex]$$
(z - 10)^2 = z^2 - 20z + 100.
$$[/tex]
Thus, we can write:
[tex]$$
z^2 - 20z = (z - 10)^2 - 100,
$$[/tex]
and so
[tex]$$
20z - z^2 = -\left[(z - 10)^2 - 100\right] = 100 - (z - 10)^2.
$$[/tex]
Step 4. Combine the rewritten parts.
Substitute the rewritten forms into the original expression:
[tex]\[
\begin{aligned}
625y^2 + 400y - 36 + 20z - z^2 &= \Bigl[(25y + 8)^2 - 100\Bigr] + \Bigl[100 - (z-10)^2\Bigr] \\
&= (25y + 8)^2 - (z-10)^2.
\end{aligned}
\][/tex]
Now the expression is written as a difference of two squares.
Step 5. Factor the difference of squares.
Recall that a difference of two squares factors as:
[tex]$$
A^2 - B^2 = (A - B)(A + B).
$$[/tex]
Here, let:
[tex]$$
A = 25y + 8 \quad \text{and} \quad B = z - 10.
$$[/tex]
Thus, the expression factors as:
[tex]\[
\begin{aligned}
(25y + 8)^2 - (z-10)^2 &= \Bigl[(25y + 8) - (z-10)\Bigr]\, \Bigl[(25y + 8) + (z-10)\Bigr] \\
&= (25y + 8 - z + 10)(25y + 8 + z - 10) \\
&= (25y - z + 18)(25y + z - 2).
\end{aligned}
\][/tex]
Final Answer:
The factored form of the expression is
[tex]$$
\boxed{(25y - z + 18)(25y + z - 2)}.
$$[/tex]
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Rewritten by : Jeany
