Write the expression in factored form: n^2 - 9n +14 = 0; n^2 + 4n - 12 =0

\( 2.)\\ \\ n^2 - 9n +14 = 0 \\ \\ a=1, \ b = -9, \ c = 14 \\ \\\Delta =b^2-4ac = (-9)^2 -4\cdot1\cdot 14 = 81 -56 = 25 \\ \\n_{1}=\frac{-b-\sqrt{\Delta} }{2a}=\frac{9-\sqrt{25}}{2 }=\frac{ 9-5}{2}=\frac{4}{2}= 2 \)

\( n_{2}=\frac{-b+\sqrt{\Delta} }{2a}=\frac{9+\sqrt{25}}{2 }=\frac{ 9+5}{2}=\frac{14}{2}= 7 \\ \\ Answer : \ n^2 - 9n +14 =(n-2)(n-7) \)



\( 3.)\\ \\ n^2 + 4n - 12 =0 \\ \\ a=1, \ b = 4, \ c = -12 \\ \\\Delta =b^2-4ac = 4^2 -4\cdot1\cdot (-12) = 16+48=64 \\ \\n_{1}=\frac{-b-\sqrt{\Delta} }{2a}=\frac{-4-\sqrt{64}}{2 }=\frac{ -4-8}{2}=\frac{-12}{2}= -6 \)

\( n_{2}=\frac{-b+\sqrt{\Delta} }{2a}= \frac{-4+\sqrt{64}}{2 }=\frac{ -4+8}{2}=\frac{4}{2}=2\\ \\ Answer : \ n^2 +4n -12 =(n+6)(n-2) \)


second way :

\( 2.)\\ \\ n^2 - 9n +14 = n^2 - 9n+2n -2n +14 = n^2 - 7x -2n +14 =\\ \\ = n(n-7)-2(n-7)=(n-7)(n- 2)\\ \\ \\ Answer : \ n^2 - 9n +14 =(n-2)(n-7) \)
 

\( 3.)\\ \\ n^2 + 4n - 12 = n^2 + 4n +2n - 2n - 12 = n^2 + 6n - 2n - 12 =\\ \\ = n(n+6) -2(n+6)=(n+6)(n-2) \\ \\ \\ Answer : \ n^2 +4n -12 =(n+6)(n-2) \)



N^2-9n +14
n^2-7n-2n+14
n(n-7) -2(n-7)
(n-2)(n-7)


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