a) Laske lausekkeen \(x^2+1\) arvo, kun \(x=2\).
b) Laske lausekkeen \(x^2+1\) arvo, kun \(x=-2\).
c) Päättele mitkä luvut sopivat sulkujen sisälle: \(\Big(\boxed{\phantom{X}}\Big)^2+1=5\).
d) Ratkaise yhtälö \(x^2+1=5\).
e) Päättele mitkä luvut sopivat sulkujen sisälle: \(\Big(\boxed{\phantom{X}}\Big)^2=16\).
f) Ratkaise yhtälö \(x^2-1=8\).
\( \require{color} \begin{align*} \textbf{a)}\qquad &x^2+1 \quad || x = 2 \\ =\ &2^2+1 \\ =&4+1\\ =\ &\underline{\underline{5}} \qquad \qquad \color{red}{\text{(+1p)}} \end{align*}\)
\( \require{color} \begin{align*} \textbf{b)}\qquad &x^2+1 \quad || x = -2 \\ =\ &(-2)^2+1 \\ =&4+1\\ =\ &\underline{\underline{5}} \qquad \qquad \color{red}{\text{(+1p)}} \end{align*}\)
\( \require{color} \begin{align*} \textbf{c) } &( \ )^2+1=5 \\ &\underline{\underline{(2) }}^2+1=5 \text{ ja } (\underline{\underline{-2}})^2+1=5 \quad \color{red}{\text{(+1p)}} \end{align*}\)
\( \require{color} \begin{align*} \textbf{d) } x^2+1&=5 \ \ \ \qquad || -1 \\ x^2&=4 \\ x&= \pm 2 \end{align*}\)
Vastaus: \(x=2\) tai \(x=-2\). \(\require{color}\color{red}{\text{(+1p)}}\)
\( \require{color} \begin{align*} \textbf{e) } &( \ )^2=16 \\ &\underline{\underline{(4) }}^2=16 \text{ ja } (\underline{\underline{-4}})^2=16 \quad \color{red}{\text{(+1p)}} \end{align*}\)
\( \require{color} \begin{align*} \textbf{f) } x^2-1&=8 \ \ \ \qquad || +1 \\ x^2&=9 \\ x&= \pm 3 \end{align*}\)
Vastaus: \(x=3\) tai \(x=-3\). \(\require{color}\color{red}{\text{(+1p)}}\)
Ratkaise yhtälö
a) \(x^2-6x=0\)
b) \(3x^2=-9x\)
\( \require{color} \begin{align*} \textbf{a)}\qquad x^2-6x&=0 \\ x\cdot x - 6 \cdot x&=0 \\ x(x-6)&=0 &&||\text{ TNS = tulon nollasääntö} \\ & &&\color{red}{\text{(+1p TNS tai ratkaisukaava)}}\\ x=0 \text{ tai } &x-6=0 &&\color{red}{\text{(+1p)}} \\ x=0 \text{ tai } &x=6 \end{align*}\)
Vastaus: \(x=0\) tai \(x=6\). \(\require{color}\color{red}{\text{(+1p)}}\)
\( \require{color} \begin{align*} \textbf{b)}\qquad\qquad 3x^2&=-9x &&||\color{blue}+9x \\ 3x^2\color{blue}+9x&=-9x\color{blue}+9x \\ 3x^2+9x&=0 \\ x \cdot 3x + x \cdot 9&=0 \\ x(3x+9)&=0 &&||\text{ TNS} \\ & &&\color{red}{\text{(+1p TNS tai ratkaisukaava)}}\\ x=0 \text{ tai } &3x+9=0 && \color{red}{\text{(+1p)}}\\ x=0 \text{ tai } &3x=-9 \\ x=0 \text{ tai } &x=-3 \end{align*}\)
Vastaus: \(x=0\) tai \(x=-3\). \(\require{color}\color{red}{\text{(+1p)}}\)
Ratkaise yhtälö
a) \(x^2+x-6=0\)
b) \(2x^2=x+1\)
\( \require{color} \begin{align*} \textbf{a) }\ &x^2+x-6=0 & &\Big|\Big|x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}\\ & && \Big|\Big| a=1, b=1, c=-6 \end{align*}\)
\( \require{color} \begin{align*} x&=\dfrac{-1 \pm \sqrt{1^2 -4 \cdot 1 \cdot (-6)}}{2 \cdot 1} \qquad \color{red} \text{(+1p)} \\ x&=\dfrac{-1 \pm \sqrt{25}}{2} \\ x&=\dfrac{-1 \pm 5}{2} \qquad\qquad\qquad\qquad\quad \ \color{red} \text{(+1p)}\\ x&=\underline{\underline{-3}} \text{ tai } x=\underline{\underline{\ 2\ }} \qquad \ \ \qquad \qquad \color{red}{\text{(+1p)}} \end{align*}\)
\( \require{color} \begin{align*} \textbf{b) }2x^2&=x+1 &&|| -x -1 \\ 2x^2&-x-1=0 &&\Big|\Big|x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a} \\ \ & &&\Big|\Big|a=2, b=-1, c=-1 \end{align*}\)
\( \require{color} \begin{align*} x&=\dfrac{-(-1) \pm \sqrt{(-1)^2-4 \cdot 2 \cdot (-1)}}{2 \cdot 2} && \color{red} \text{(+1p)} &&& &&&& &&&&& \\ x&=\dfrac{1 \pm \sqrt{9}}{4} \\ x&=\dfrac{1 \pm 3}{4} && \color{red} \text{(+1p)}\\ x&=\underline{\underline{\ 1\ }} \text{ tai } x=\underline{\underline{-\dfrac{1}{2}}} && \color{red} \text{(+1p)} \end{align*}\)
Tekemäsi itsearvion pohjalta tuloksesi prosentteina on: