7 Given that $$ m+n=9 $$ and $$ m n=8 $$, find the value of $$ \begin{array}{ll} ext { (a) } m^{2}+n^{2}, & ext { (b) }(m-n)^{2}\end{array} $$

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7 Given that $$ m+n=9 $$ and $$ m n=8 $$, find the value of $$ \begin{array}{ll}	ext { (a) } m^{2}+n^{2}, & 	ext { (b) }(m-n)^{2}\end{array} $$

UpStudy AI · Accepted Answer

We are given:
- $$ m+n=9 $$
- $$ mn=8 $$

**Step 1: Calculate $$ m^2+n^2 $$**
  
Recall the identity:
$$
(m+n)^2 = m^2 + n^2 + 2mn
$$
Substitute the known values:
$$
9^2 = m^2+n^2+2(8)
$$
This simplifies to:
$$
81 = m^2+n^2+16
$$
Solve for $$ m^2+n^2 $$:
$$
m^2+n^2 = 81-16=65
$$
  
**Step 2: Calculate $$ (m-n)^2 $$**
  
Recall the identity:
$$
(m-n)^2 = m^2+n^2-2mn
$$
We already found $$ m^2+n^2=65 $$ and we know $$ mn=8 $$. Substitute these values:
$$
(m-n)^2 = 65-2(8)
$$
$$
(m-n)^2 = 65-16=49
$$

**Final Answers:**

$$ m^2+n^2=65 $$

$$ (m-n)^2=49 $$
$$ m^2 + n^2 = 65 $$ $$ (m - n)^2 = 49 $$

7 Given that \( m+n=9 \) and \( m n=8 \), find the value of \( \begin{array}{ll}\text { (a) } m^{2}+n^{2}, & \text { (b) }(m-n)^{2}\end{array} \)

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