For å faktorisere et andregradsuttrykk kan vi løse likningen med abc-formelen og bruke svarene inn i det generelle uttrykket

\begin{align}a\cdot x^2+b\cdot x+c &=0\\ x &=\frac{-b\pm\sqrt{b^2-4\cdot a\cdot c}}{2\cdot a}\\
x=x_1 &\vee x= x_2\\a\cdot x^2+b\cdot x+c&=a(x-x_1)(x-x_2) \end{align}

Oppgave 1

  • Løsningsforslag:
    \begin{align}x^2+4x+3&=0\\x&=\frac{-4\pm\sqrt{16-12}}{2}\\x&=\frac{-4\pm2}{2}\\ x=\frac{-4+2}{2}&\vee x=\frac{-4-2}{2}\\x=-1 &\vee x=-3\\ \\
    x^2+4x+3&=(x+3)(x+1)
    \end{align}

  • Løsningsforslag:

    \begin{align}
    x^2+6x+8&=0\\
    x&=\frac{-6\pm\sqrt{36-32}}{2}\\
    x&=\frac{-6\pm2}{2}\\
    x=\frac{-6 + 2}{2}
    &\vee x=\frac{-6-2}{2}\\
    x=-2 &\vee x=-4\\ \\
    x^2+6x+8&=(x+2)(x+4)
    \end{align}

  • Løsningsforslag:


    \begin{align}
    x^2+8x+15&=0\\
    x&=\frac{-8\pm\sqrt{8^2-4\cdot 15}}{2}\\
    &=\frac{-8\pm\sqrt{64-60}}{2}\\
    &=\frac{-8\pm\sqrt{4}}{2}\\
    &=\frac{-8\pm 2}{2}\\
    x=\frac{-8 - 2}{2}&\vee x=\frac{-8 + 2}{2}\\
    x=\frac{-10}{2} &\vee x=\frac{-6}{2}\\
    x=-5 &\vee x=-3\\ \\
    x^2+8x+15&=(x+3)(x+5)
    \end{align}

  • Løsningsforslag:
    \begin{align}
    x^2+10x+24&=0\\
    x&=\frac{-10\pm\sqrt{100-96}}{2}\\
    x&=\frac{-10\pm 2}{2}\\
    x=\frac{-8}{2} &\vee x=\frac{-12}{2}\\
    x=-4 &\vee x=-6\\ \\
    x^2+10x+24&=(x+4)(x+6)\\
    \end{align}

  • Løsningsforslag:
    \begin{align}x^2+12x+35&=0\\
    x&=\frac{-12\pm\sqrt{12^2-4\cdot 1\cdot 35}}{2\cdot 1}\\
    x&=\frac{-12\pm \sqrt{144-140}}{2}\\
    x&=\frac{-12\pm \sqrt{4}}{2}\\
    x&=\frac{-12\pm 2}{2}\\
    x=\frac{-12+2}{2} &\vee x=\frac{-12-2}{2}\\
    x= -5 &\vee x=-7\\ \\
    x^2+12x+35&=(x+7)(x+5)
    \end{align}

  • \begin{align}
    x^2+14x+48&=(x+6)(x+8)
    \end{align}

    \begin{align}
    x^2+14x+48&=0\\
    x&=\frac{-14\pm\sqrt{14^2-4\cdot 1\cdot 48}}{2}\\
    &=\frac{-14\pm\sqrt{196-192}}{2}\\
    &=\frac{-14\pm\sqrt{4}}{2}\\
    &=\frac{-14\pm2}{2}\\
    \frac{-14+2}{2}&\vee x=\frac{-14-2}{2}\\
    x=\frac{-12}{2}&\vee x=\frac{-16}{2}\\
    x=-6 &\vee x=-8\\
    \end{align}

  • \begin{align}
    x^2+16x+63&=(x+9)(x+7)\\
    x^2+16x+63&=0\\
    x&=\frac{-16\pm \sqrt{16^2-4\cdot 63}}{2}\\
    x&=\frac{-16\pm \sqrt{256-252}}{2}\\
    x&=\frac{-16\pm 2}{2}\\
    x=\frac{-16+2}{2} &\vee x=\frac{-16-2}{2}\\
    x=-7 &\vee x=-9
    \end{align}

  • \begin{align}x^2+10x+16=(x+2)(x+8)\end{align}

  • \begin{align}x^2+12x+27=(x+3)(x+9)\end{align}

  • \begin{align}x^2+12x+20=(x+2)(x+10)\end{align}

Oppgave 2

  • \begin{align}
    x^2-2x-3&=0\\
    x&=\frac{-(-2)\pm\sqrt{(-2)^2-4\cdot(-3)}}{2}\\
    x&=\frac{2\pm\sqrt{4+12}}{2}\\
    x&=\frac{2\pm 4}{2}\\
    x=\frac{6}{2}&\vee x=\frac{-2}{2}\\
    x=3 &\vee x=-1\\ \\
    x^2-2x-3&=(x-3)(x+1)\\
    \end{align}

  • \begin{align}
    x^2-2x-8&=0\\
    x&=\frac{-(-2)\pm\sqrt{(-2)^2-4\cdot (-8)}}{2}\\
    &=\frac{2\pm\sqrt{4+32}}{2}\\
    &=\frac{2\pm\sqrt{36}}{2}\\
    &=\frac{2\pm 6}{2}\\
    x=\frac{2+6}{2} &\vee x=\frac{2-6}{2}\\
    x=4 &\vee x=-2\\ \\
    x^2-2x-8&=(x-4)(x+2)
    \end{align}

  • \begin{align}
    x^2-2x-15&=0\\
    x&=\frac{-(-2)\pm\sqrt{(-2)^2-4\cdot (-15)}}{2}\\
    &=\frac{2\pm\sqrt{4+60}}{2}\\
    &=\frac{2\pm\sqrt{64}}{2}\\
    &=\frac{2\pm 8}{2}\\
    x=\frac{10}{2} &\vee x=\frac{-6}{2}\\
    x=5 &\vee x=-3\\ \\
    x^2-2x-15&=(x-5)(x+3)
    \end{align}

  • \begin{align}
    x^2-2x-24&=0\\
    x&=\frac{-(-2)\pm\sqrt{(-2)^2-4\cdot(-24)}}{2}\\
    &=\frac{2\pm\sqrt{4+96}{2}\\
    &=\frac{2\pm 10}{2}\\
    x= \frac{12}{2}&\vee x=\frac{-8}{2}\\
    x=6 &\vee x=-4\\ \\
    x^2-2x-24&=(x-6)(x+4)
    \end{align}

  • \begin{align}
    x^2-2x-35&=0\\
    x&=\frac{2\pm\sqrt{4+140}}{2}\\
    &=\frac{2\pm\sqrt{144}}{2}\\
    &=\frac{2\pm 12}{2}\\
    x=\frac{14}{2} &\vee x=\frac{-10}{2}\\
    x=7 &\vee x=-5\\ \\
    x^2-2x-35&=(x-7)(x+5)
    \end{align}

Oppgave 3

  • \begin{align}x^2-2x-48=(x-8)(x+6)\end{align}

  • \begin{align}x^2-2x-63=(x-9)(x+7)\end{align}

  • \begin{align}x^2+6x-16=(x+8)(x-2)\end{align}

  • \begin{align}x^2+6x-27=(x+9)(x-3)\end{align}

  • \begin{align}x^2+8x-20=(x+10)(x-2)\end{align}

Oppgave 4

  • \begin{align}
    x^2-4x+3&=0\\
    x&=\frac{4\pm\sqrt{16-12}}{2}\\
    &=\frac{4\pm 2}{2}\\
    x=\frac{6}{2} &\vee x=\frac{2}{2}\\
    x=3 &\vee 1\\ \\
    x^2-4x+3&=(x-3)(x-1)
    \end{align}

  • \begin{align}
    x^2-6x+8 &=0\\
    x&=\frac{-(-6)\pm\sqrt{36-32}}{2}\\
    x^2-6x+8 &=(x-2)(x-4)
    \end{align}

  • \begin{align}x^2-8x+15=(x-3)(x-5)\end{align}

  • \begin{align}x^2-10x+24=(x-6)(x-4)\end{align}

  • \begin{align}x^2-12x+35=(x-5)(x-7)\end{align}

Oppgave 5

  • \begin{align}x^2-14x+48=(x-8)(x-6)\end{align}

  • \begin{align}x^2-16x+63=(x-7)(x-9)\end{align}

  • \begin{align}x^2-10x+16=(x-2)(x-8)\end{align}

  • \begin{align}x^2-12x+27=(x-9)(x-3)\end{align}

  • \begin{align}x^2-12x+20=(x-2)(x-10)\end{align}

Oppgave 6