1. kvadratsetning:
(\textcolor{red}{a} + \textcolor{blue}{b})^2 = \textcolor{red}{a}^2 + 2 \textcolor{red}{a} \textcolor{blue}{b} + \textcolor{blue}{b}^22. kvadratsetning:
(\textcolor{red}{a} - \textcolor{blue}{b})^2 = \textcolor{red}{a}^2 - 2 \textcolor{red}{a} \textcolor{blue}{b} + \textcolor{blue}{b}^23. kvadratsetning:
(\textcolor{red}{a} + \textcolor{blue}{b}) (\textcolor{red}{a} - \textcolor{blue}{b})
= \textcolor{red}{a}^2 - \textcolor{blue}{b}^2+ Bevis for 1. kvadratsetning
Begynn med å dele opp den ene parentesen:
\begin{aligned}
(a+b)^2 &= (\textcolor{green}{a}+ \textcolor{purple}{b})(a+b) \\
& = \textcolor{green}{a}(a+b) + \textcolor{purple}{b}(a+b) \\
& = (\textcolor{green}{a}a + \textcolor{green}{a} b) + (\textcolor{purple}{b} a + \textcolor{purple}{b}b) \\
& = \textcolor{green}{a}a + \textcolor{green}{a} b + \textcolor{purple}{b} a + \textcolor{purple}{b}b \\
&= a^2 + 2ab + b^2
\end{aligned}+ Bevis for 2. kvadratsetning
Begynn med å dele opp den ene parentesen:
\begin{aligned}
(a-b)^2 &= (\textcolor{green}{a}- \textcolor{purple}{b})(a-b) \\
& = \textcolor{green}{a}(a-b) - \textcolor{purple}{b}(a-b) \\
& = (\textcolor{green}{a}a - \textcolor{green}{a} b) - (\textcolor{purple}{b} a - \textcolor{purple}{b}b) \\
& = \textcolor{green}{a}a - \textcolor{green}{a} b - \textcolor{purple}{b} a + \textcolor{purple}{b}b \\
&= a^2 - 2ab + b^2
\end{aligned}+ Bevis for 3. kvadratsetning
Begynn med å dele opp den første parentesen:
\begin{aligned}
(\textcolor{green}{a}+ \textcolor{purple}{b})(a-b)
& = \textcolor{green}{a}(a-b) + \textcolor{purple}{b}(a-b) \\
& = \textcolor{green}{a}a - \textcolor{green}{a} b + \textcolor{purple}{b} a - \textcolor{purple}{b}b \\
&= a^2 - b^2
\end{aligned}+ Eksempel 1: $(x+3)^2$
Her kan vi bruke 1. kvadratsetning:
\begin{aligned}
\textnormal{Regel: } \quad & (\textcolor{red}{a} + \textcolor{blue}{b})^2 = \textcolor{red}{a}^2 + 2 \textcolor{red}{a} \textcolor{blue}{b} + \textcolor{blue}{b}^2 \\
\Rightarrow \quad
& (\textcolor{red}{x} + \textcolor{blue}{3})^2 = \textcolor{red}{x}^2 + 2 \cdot \textcolor{red}{x} \cdot \textcolor{blue}{3} + \textcolor{blue}{3}^2 \\
\Rightarrow \quad &(x + 3)^2 = x^2 + 6x + 9
\end{aligned}+ Eksempel 2: $(x-3)^2$
Her kan vi bruke 2. kvadratsetning:
\begin{aligned}
\textnormal{Regel: } \quad & (\textcolor{red}{a} - \textcolor{blue}{b})^2 = \textcolor{red}{a}^2 - 2 \textcolor{red}{a} \textcolor{blue}{b} + \textcolor{blue}{b}^2 \\
\Rightarrow \quad
& (\textcolor{red}{x} - \textcolor{blue}{3})^2 = \textcolor{red}{x}^2 - 2 \cdot \textcolor{red}{x} \cdot \textcolor{blue}{3} + \textcolor{blue}{3}^2 \\
\Rightarrow \quad &(x - 3)^2 = x^2 - 6x + 9
\end{aligned}+ Eksempel 3: $(x+3)(x-3)$
Her kan vi bruke 3. kvadratsetning:
\begin{aligned}
\textnormal{Regel: } \quad & (\textcolor{red}{a} + \textcolor{blue}{b}) (\textcolor{red}{a} - \textcolor{blue}{b})
= \textcolor{red}{a}^2 - \textcolor{blue}{b}^2 \\
\Rightarrow \quad & (\textcolor{red}{x} + \textcolor{blue}{3}) (\textcolor{red}{x} - \textcolor{blue}{3})
= \textcolor{red}{x}^2 - \textcolor{blue}{3}^2 \\
\Rightarrow \quad & (\textcolor{red}{x} + \textcolor{blue}{3}) (\textcolor{red}{x} - \textcolor{blue}{3})
= x^2 - 9
\end{aligned}+ Eksempel: $(2+3)^2$
Her kan vi bruke 1. kvadratsetning:
\begin{aligned}
\textnormal{Regel: } \quad & (\textcolor{red}{a} + \textcolor{blue}{b})^2 = \textcolor{red}{a}^2 + 2 \textcolor{red}{a} \textcolor{blue}{b} + \textcolor{blue}{b}^2 \\
\Rightarrow \quad
& (\textcolor{red}{2} + \textcolor{blue}{3})^2 = \textcolor{red}{2}^2 + 2 \cdot \textcolor{red}{2} \cdot \textcolor{blue}{3} + \textcolor{blue}{3}^2 \\
\Rightarrow \quad &(2 + 3)^2 = 4 + 12 + 9 = 25
\end{aligned}Men her er det selvsagt mye enklere å regne ut parentesen først:
(2+3)^2 = 5^2 = 25
+ Hoderegningstriks: $21^2$
Hvis du vil regne ut $21^2$ i hodet, kan du bruke 1. kvadratsetning:
\begin{aligned}
\textnormal{Regel: } \quad & (\textcolor{red}{a} + \textcolor{blue}{b})^2 = \textcolor{red}{a}^2 + 2 \textcolor{red}{a} \textcolor{blue}{b} + \textcolor{blue}{b}^2 \\
\Rightarrow \quad
& (\textcolor{red}{20} + \textcolor{blue}{1})^2 = \textcolor{red}{20}^2 + 2 \cdot \textcolor{red}{20} \cdot \textcolor{blue}{1} + \textcolor{blue}{1}^2 \\
\Rightarrow \quad & 21^2 = 20^2 + 20 + 21 = 441
\end{aligned}Ser du systemet?
\textcolor{red}{\textnormal{Tall}}^2 = (\textcolor{blue}{\textnormal{Tall} - 1})^2 + (\textcolor{blue}{\textnormal{Tall} - 1}) + \textcolor{red}{\textnormal{Tall}}La oss prøve med noen flere tall:
\begin{aligned}
\textcolor{red}{6}^2 &= \textcolor{blue}{5}^2 + \textcolor{blue}{5} + \textcolor{red}{6} = 25 + 11 = 36 \\
\textcolor{red}{11}^2 &= \textcolor{blue}{10}^2 + \textcolor{blue}{10} + \textcolor{red}{11} = 100 + 21 = 121 \\
\textcolor{red}{41}^2 &= \textcolor{blue}{40}^2 + \textcolor{blue}{40} + \textcolor{red}{41} = 1600 + 81 = 1681 \\
\textcolor{red}{101}^2 &= \textcolor{blue}{100}^2 + \textcolor{blue}{100} + \textcolor{red}{101} = 10\;000 + 201 = 10\;201
\end{aligned}Du kan selvsagt bruke samme metode for andre tall enn en over noe du kjenner:
\begin{aligned}
\textnormal{Regel: } \quad & (\textcolor{red}{a} + \textcolor{blue}{b})^2 = \textcolor{red}{a}^2 + 2 \textcolor{red}{a} \textcolor{blue}{b} + \textcolor{blue}{b}^2 \\
\Rightarrow \quad
& (\textcolor{red}{20} + \textcolor{blue}{6})^2 = \textcolor{red}{20}^2 + 2 \cdot \textcolor{red}{20} \cdot \textcolor{blue}{6} + \textcolor{blue}{6}^2 \\
\Rightarrow \quad & 26^2 = 20^2 + 240 + 36 = 676
\end{aligned}Og, vips, kan du imponere med litt hoderegning.
+ Hoderegningstriks: $25^2$
Hvis du vil regne ut $25^2$ i hodet, kan du bruke 3. kvadratsetning:
\begin{aligned}
\textnormal{Regel: } \quad & (\textcolor{red}{a} + \textcolor{blue}{b}) (\textcolor{red}{a} - \textcolor{blue}{b}) = \textcolor{red}{a}^2 - \textcolor{blue}{b}^2 \\
\Rightarrow \quad
& (\textcolor{red}{25} + \textcolor{blue}{5}) (\textcolor{red}{25} - \textcolor{blue}{5}) = \textcolor{red}{25}^2 - \textcolor{blue}{5}^2 \\
\Rightarrow \quad & \textcolor{red}{25}^2 = (\textcolor{red}{25} + \textcolor{blue}{5}) (\textcolor{red}{25} - \textcolor{blue}{5}) + \textcolor{blue}{5}^2 \\
\Rightarrow \quad & 25^2 = 30 \cdot 20 + 25 = 625
\end{aligned}Ser du systemet?
\textcolor{red}{\textnormal{Tall}}^2 = (\textcolor{blue}{\textnormal{Tall} + 5}) (\textcolor{blue}{\textnormal{Tall} - 5}) + \textcolor{red}{5}^2La oss prøve med noen flere tall:
\begin{aligned}
\textcolor{red}{15}^2 &= \textcolor{blue}{20} \cdot \textcolor{blue}{10} + \textcolor{red}{25} = 200 + 25 = 225 \\
\textcolor{red}{35}^2 &= \textcolor{blue}{40} \cdot \textcolor{blue}{30} + \textcolor{red}{25} = 1200 + 25 = 1225 \\
\textcolor{red}{55}^2 &= \textcolor{blue}{60} \cdot \textcolor{blue}{50} + \textcolor{red}{25} = 3000 + 25 = 3025 \\
\textcolor{red}{105}^2 &= \textcolor{blue}{110} \cdot \textcolor{blue}{100} + \textcolor{red}{25} = 11\;000 + 25 = 11\;025 \\
\end{aligned}Og, vips, kan du imponere med litt hoderegning.