https://peda.net/id/56294efc5f0 2020-03-05T19:04:12+02:00 https://peda.net/:static/535/peda.net.logo.bg.svg <div class="license">Tämän sivun lisenssi <a rel="license" href="https://peda.net/info">Peda.net-yleislisenssi</a></div> https://peda.net/id/4e507fe661d 2020-03-09T09:48:03+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E3%2B2x%5E2%2B5x%3D0" alt="x^3+2x^2+5x=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5Cleft(x%5E2%2B2x%2B5%5Cright)%3D0" alt="x\left(x^2+2x+5\right)=0"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D0%5C%20tai%5C%20x%5E2%2B2x%2B5%3D0" alt="x=0\ tai\ x^2+2x+5=0"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D0%5C%20tai%5C%20x%3D-1%5Cpm2i" alt="x=0\ tai\ x=-1\pm2i"/><br/> b)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E3%2Bx%5E2%2B2x-4%3D0" alt="x^3+x^2+2x-4=0"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-1%5Cpm%5Csqrt%7B3%7D%5Ccdot%20i%5C%20tai%5C%20x%3D1" alt="x=-1\pm\sqrt{3}\cdot i\ tai\ x=1"/></div> <div> </div> </div> 2020-03-09T09:48:03+02:00 https://peda.net/id/10a898c861d 2020-03-09T09:39:10+02:00 <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E3-5x%3D-2x%5E2%2B6" alt="x^3-5x=-2x^2+6"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E3%2B2x%5E2-5x-6%3D0" alt="x^3+2x^2-5x-6=0"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cpm1%7B%2C%7D%5C%20%5Cpm2%7B%2C%7D%5C%20%5Cpm3%7B%2C%7D%5C%20%5Cpm6" alt="\pm1{,}\ \pm2{,}\ \pm3{,}\ \pm6"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D2%7B%2C%7D%5C%20x%3D-1%5C%20tai%5C%20x%3D-3" alt="x=2{,}\ x=-1\ tai\ x=-3"/></div> 2020-03-09T09:39:10+02:00 https://peda.net/id/5baaf09c61d 2020-03-09T09:34:06+02:00 a) 3<br/> b) 1, 2, 4, 8<br/> c) -1, 2, 4<br/> tarkistus:<br/> <img src="https://i.imgur.com/fBuPU3r.png" alt=""/> 2020-03-09T09:34:06+02:00 https://peda.net/id/f69bf50e61d 2020-03-09T09:52:45+02:00 funktio on kasvava väleillä <img src="https://math-demo.abitti.fi/math.svg?latex=-1%3Cx%3C2%5C%20ja%5C%202%3Cx" alt="-1&lt;x&lt;2\ ja\ 2&lt;x"/> 2020-03-09T09:52:45+02:00 https://peda.net/id/c9b3932461d 2020-03-09T09:30:01+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E4%3D3x%5E2" alt="x^4=3x^2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E4-3x%5E2%3D0" alt="x^4-3x^2=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%5Cleft(x%5E2-3%5Cright)%3D0" alt="x^2\left(x^2-3\right)=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%3D0%5C%20tai%5C%20x%5E2-3%3D0" alt="x^2=0\ tai\ x^2-3=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D0%5C%20tai%5C%20x%3D%5Csqrt%7B3%7D" alt="x=0\ tai\ x=\sqrt{3}"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2x%5E3%2B3x%5E2%3D1" alt="2x^3+3x^2=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%5Cleft(2x%2B3%5Cright)%3D1" alt="x^2\left(2x+3\right)=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-1%5C%20tai%5C%20x%3D%5Cfrac%7B1%7D%7B2%7D" alt="x=-1\ tai\ x=\frac{1}{2}"/> 2020-03-09T09:30:01+02:00 https://peda.net/id/d618af8861d 2020-03-09T09:23:13+02:00 <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Cfrac%7B3x%5E3-11x%5E2-6x%2B8%7D%7Bx-4%7D" alt="f\left(x\right)=\frac{3x^3-11x^2-6x+8}{x-4}"/><br/> <br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bmatrix%7D%0A%263x%5E2%2Bx-2%5C%5C%0Ax-4%263x%5E3-11x%5E2-6x%2B8%5C%5C%0A%263x%5E3%2B12x%5E2%5C%5C%0A%26--------%5C%5C%0A%26x%5E2-6x%5C%5C%0A%26x%5E2-4x%5C%5C%0A%26--------%5C%5C%0A%26-2x%2B8%5C%5C%0A%26-2x%2B8%5C%5C%0A%26--------%5C%5C%0A%260%0A%5Cend%7Bmatrix%7D" alt="\begin{matrix}&#10;&amp;3x^2+x-2\\&#10;x-4&amp;3x^3-11x^2-6x+8\\&#10;&amp;3x^3+12x^2\\&#10;&amp;--------\\&#10;&amp;x^2-6x\\&#10;&amp;x^2-4x\\&#10;&amp;--------\\&#10;&amp;-2x+8\\&#10;&amp;-2x+8\\&#10;&amp;--------\\&#10;&amp;0&#10;\end{matrix}"/></div> <div>eli</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(x-7%5Cright)%5Cleft(3x%5E2%2Bx-2%5Cright)%3D0" alt="\left(x-7\right)\left(3x^2+x-2\right)=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x-7%3D0%5C%20tai%5C%203x%5E2%2Bx-2%3D0" alt="x-7=0\ tai\ 3x^2+x-2=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D7" alt="x=7"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=tai" alt="tai"/><!--filtered attribute: style="display: inline;"--><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B-1%5Cpm%5Csqrt%7B1%5E2%2B24%7D%7D%7B6%7D%3D%5Cfrac%7B-1%5Cpm5%7D%7B6%7D%3D-1%5C%20tai%5C%20%5Cfrac%7B2%7D%7B3%7D" alt="x=\frac{-1\pm\sqrt{1^2+24}}{6}=\frac{-1\pm5}{6}=-1\ tai\ \frac{2}{3}"/></div> <div> </div> <div> </div> 2020-03-09T09:23:13+02:00 https://peda.net/id/7e271ca661d 2020-03-09T08:55:23+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-x%5E2-2x-2%3D0" alt="-x^2-2x-2=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B2%5Cpm%5Csqrt%7B4-4%5Ccdot%5Cleft(-1%5Cright)%5Ccdot%5Cleft(-2%5Cright)%7D%7D%7B-1%5Ccdot2%7D%3D%5Cfrac%7B2%5Cpm%5Csqrt%7B4-8%7D%7D%7B-2%7D%3D%5Cfrac%7B2%5Cpm%5Csqrt%7B4%7D%5Ccdot%5Csqrt%7B-1%7D%7D%7B-2%7D%3D%5Cfrac%7B2%5Cpm2i%7D%7B-2%7D%3D-1%5Cpm%20i" alt="x=\frac{2\pm\sqrt{4-4\cdot\left(-1\right)\cdot\left(-2\right)}}{-1\cdot2}=\frac{2\pm\sqrt{4-8}}{-2}=\frac{2\pm\sqrt{4}\cdot\sqrt{-1}}{-2}=\frac{2\pm2i}{-2}=-1\pm i"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E4%2Bx%5E2%3D0" alt="x^4+x^2=0"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%5Cleft(x%5E2%2B1%5Cright)%3D0" alt="x^2\left(x^2+1\right)=0"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%3D0%5C%20tai%5C%20x%5E2%2B1%3D0" alt="x^2=0\ tai\ x^2+1=0"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D0%5C%20tai%5C%20x%3D%5Cpm%20i" alt="x=0\ tai\ x=\pm i"/><br/> c)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(3%2B2i%5Cright)%2B%5Cleft(2-i%5Cright)%3D5%2Bi" alt="\left(3+2i\right)+\left(2-i\right)=5+i"/><br/> d)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(3%2B2i%5Cright)%5Ccdot%5Cleft(2-i%5Cright)%3D4%2Bi" alt="\left(3+2i\right)\cdot\left(2-i\right)=4+i"/></div> 2020-03-09T08:44:58+02:00