a) Laske lausekkeen \(x^2+3\) arvo, kun \(x=4\).
b) Laske lausekkeen \(x^2+3\) arvo, kun \(x=-4\).
c) Päättele mitkä luvut sopivat sulkujen sisälle: \(\Big(\boxed{\phantom{X}}\Big)^2+3=19\).
d) Ratkaise yhtälö \(x^2+3=19\).
e) Päättele mitkä luvut sopivat sulkujen sisälle: \(\Big(\boxed{\phantom{X}}\Big)^2=36\).
f) Ratkaise yhtälö \(x^2-5=4\).
\( \require{color} \begin{align*} \textbf{a)}\qquad &x^2+3 \quad || x = 4 \\ =\ &4^2+3 \\ =&16+3\\ =\ &\underline{\underline{19}} \qquad \qquad \color{red}{\text{(+1p)}} \end{align*}\)
\( \require{color} \begin{align*} \textbf{b)}\qquad &x^2+3 \quad || x = -4 \\ =\ &(-4)^2+3 \\ =&16+3\\ =\ &\underline{\underline{19}} \qquad \qquad \color{red}{\text{(+1p)}} \end{align*}\)
\( \require{color} \begin{align*} \textbf{c) } &( \ )^2+3=19 \\ &\underline{\underline{(4) }}^2+3=19 \text{ ja } (\underline{\underline{-4}})^2+3=19 \quad \color{red}{\text{(+1p)}} \end{align*}\)
\( \require{color} \begin{align*} \textbf{d) } x^2+3&=19 \ \ \ \qquad || +1 \\ x^2&=16 \\ x&= \pm 4 \end{align*}\)
Vastaus: \(x=4\) tai \(x=-4\). \(\require{color}\color{red}{\text{(+1p)}}\)
\( \require{color} \begin{align*} \textbf{e) } &( \ )^2=36 \\ &\underline{\underline{(6) }}^2=36 \text{ ja } (\underline{\underline{-6}})^2=36 \quad \color{red}{\text{(+1p)}} \end{align*}\)
\( \require{color} \begin{align*} \textbf{f) } x^2-5&=4 \ \ \ \qquad || +5 \\ x^2&=9 \\ x&= \pm 3 \end{align*}\)
Vastaus: \(x=3\) tai \(x=-3\). \(\require{color}\color{red}{\text{(+1p)}}\)
Ratkaise yhtälö
a) \(x^2-13x=0\)
b) \(-4x^2=-10x\)
\( \require{color} \begin{align*} \textbf{a)}\qquad x^2-13x&=0 \\ x\cdot x -13 \cdot x&=0 \\ x(x-13)&=0 &&||\text{ TNS = tulon nollasääntö} \\ & &&\color{red}{\text{(+1p TNS tai ratkaisukaava)}}\\ x=0 \text{ tai } &x-13=0 &&\color{red}{\text{(+1p)}} \\ x=0 \text{ tai } &x=13 \end{align*}\)
Vastaus: \(x=0\) tai \(x=13\). \(\require{color}\color{red}{\text{(+1p)}}\)
\( \require{color} \begin{align*} \textbf{b)}\qquad\qquad -4x^2&=-10x &&||\color{blue}+10x \\ -4x^2\color{blue}+10x&=-10x\color{blue}+10x \\ -4x^2+10x&=0 \\ -4x \cdot x +10 \cdot x&=0 \\ x(-4x+10)&=0 &&||\text{ TNS} \\ & &&\color{red}{\text{(+1p TNS tai ratkaisukaava)}}\\ x=0 \text{ tai } &-4x+10=0 && \color{red}{\text{(+1p)}}\\ x=0 \text{ tai } &-4x=-10 \\ x=0 \text{ tai } &x=\dfrac{-10}{-4}=\dfrac{5}{2} \end{align*}\)
Vastaus: \(x=0\) tai \(x=\dfrac{5}{2}\). \(\require{color}\color{red}{\text{(+1p)}}\)
Ratkaise yhtälö
a) \(x^2-2x-3=0\)
b) \(2x^2=-5x+3\)
\( \require{color} \begin{align*} \textbf{a) }\ &x^2-2x-3=0 & &\Big|\Big|x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}\\ & && \Big|\Big| a=1, b=-2, c=-3 \end{align*}\)
\( \require{color} \begin{align*} x&=\dfrac{-(-2) \pm \sqrt{(-2)^2 -4 \cdot 1 \cdot (-3)}}{2 \cdot 1} \qquad \color{red} \text{(+1p)} \\ x&=\dfrac{2 \pm \sqrt{16}}{2} \\\\ x&=\dfrac{2 \pm 4}{2} \qquad\qquad\qquad\qquad\qquad\qquad \quad \color{red} \text{(+1p)}\\\\ x&=\underline{\underline{3}} \text{ tai } x=\underline{\underline{-1}} \qquad\qquad \qquad \qquad \qquad \color{red}{\text{(+1p)}} \end{align*}\)
\( \require{color} \begin{align*} \textbf{b) }2x^2&=-5x+3 &&|| +5x-3 \\ 2x^2&+5x-3=0 &&\Big|\Big|x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a} \\ \ & &&\Big|\Big|a=2, b=5, c=-3 \end{align*}\)
\( \require{color} \begin{align*} x&=\dfrac{-5 \pm \sqrt{5^2-4 \cdot 2 \cdot (-3)}}{2 \cdot 2} && \color{red} \text{(+1p)} \\ x&=\dfrac{-5 \pm \sqrt{49}}{4} \\ x&=\dfrac{-5 \pm 7}{4} && \color{red} \text{(+1p)}\\ x&=\dfrac{2}{4}=\underline{\underline{\dfrac{1}{2}}} \text{ tai } x=\dfrac{-12}{4}=\underline{\underline{-3}} && \color{red} \text{(+1p)} \end{align*}\)
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