Quadratic equation - Andregradsligning
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Forenkle uttrykket \(\(a + 2b - c\)\(a - 2b + c\)\)
Gang leddene sammen. 1.1->2.1, 1.1->2.2, 1.1->2.3, 1.2->2.1 osv...
Deretter sorter og forenkle.
\[ \require{cancel} \begin{align} (a + 2b - c)(a - 2b + c) & \quad \text{Første ledd: } a(a) + a(-2b) + a(c)\\ & \quad \text{Andre ledd: } 2b(a) + 2b(-2b) + 2b(c)\\ & \quad \text{Tredje ledd: } (-c)(a) + (-c)(-2b) + (-c)(c)\\ & \\ & \text{Leddene ganget sammen, så får vi:}\\ & = a^2 - 2ab + ac + 2ab - 4b^2 + 2bc - ac + 2bc - c^2\\ & \\ & \text{Sorter og forenkle:}\\ & = a^2 \cancel{- 2ab + 2ab} - 4b^2 \cancel{+ ac - ac} + 2bc + 2bc - c^2\\ & = a^2 - 4b^2 + 4bc - c^2 \end{align}\]
Løs/utvid uttrykket \((m^{\frac{1}{3}}+y^{-\frac{1}{2}})\)
\[ \begin{align} (m^{\frac{1}{3}}+y^{-\frac{1}{2}}) & = (m^{\frac{1}{3}} + y^{-\frac{1}{2}})(m^{\frac{1}{3}} + y^{-\frac{1}{2}})\\ & = m^{\frac{1}{3}}(m^{\frac{1}{3}} + y^{-\frac{1}{2}}) + y^{-\frac{1}{2}}(m^{\frac{1}{3}}+y^{-\frac{1}{2}})\\ & = m^{\frac{1}{3} + {\frac{1}{3}}} + m^{\frac{1}{3}}y^{-\frac{1}{2}} + m^{\frac{1}{3}}y^{-\frac{1}{2}} + y^{-\frac{1}{2}+(-\frac{1}{2})}\\ & = m^{\frac{2}{3}} + m^{\frac{1}{3}}y^{-\frac{1}{2}} + m^{\frac{1}{3}}y^{-\frac{1}{2}} + y^{-1}\\ & = m^{\frac{2}{3}} + 2m^{\frac{1}{3}}y^{-\frac{1}{2}} + y^{-1} \end{align}\]