Pregunta
7 Given that \( m+n=9 \) and \( m n=8 \), find the value of \( \begin{array}{ll}\text { (a) } m^{2}+n^{2}, & \text { (b) }(m-n)^{2}\end{array} \)
Ask by Hart Kelly. in Singapore
Mar 13,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
Responder
\( m^2 + n^2 = 65 \)
\( (m - n)^2 = 49 \)
Solución
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:**
\( m^2+n^2=65 \)
\( (m-n)^2=49 \)
Respondido por UpStudy AI y revisado por un tutor profesional
Como
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Extra Insights
To solve for \( m^2 + n^2 \) and \( (m - n)^2 \) given \( m + n = 9 \) and \( mn = 8 \), we start with the first equation. We know that: \[ m^2 + n^2 = (m+n)^2 - 2mn \] Substituting in the values: \[ m^2 + n^2 = 9^2 - 2 \cdot 8 = 81 - 16 = 65 \] So, \( m^2 + n^2 = 65 \). Next, to find \( (m - n)^2 \): \[ (m - n)^2 = (m+n)^2 - 4mn \] Substituting the known values: \[ (m - n)^2 = 9^2 - 4 \cdot 8 = 81 - 32 = 49 \] Thus, \( (m - n)^2 = 49 \). The final answers are: (a) \( m^2 + n^2 = 65 \) (b) \( (m - n)^2 = 49 \)

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