Fullstendige kvadrater er en metode for å faktorisere uttrykk med konjungatsetningen.

Oppgave 1

  • \begin{align}
    x^2+4x+3
    &=x^2+4x\color{red}{ + 4 - 4 }\color{blue} + 3\\
    &=\color{red}{x^2+4x+4}\color{blue}-4+3\\
    &=(x+2)^2-1\\&=(x+2+1)(x+2-1)\\
    &=(x+3)(x+1)
    \end{align}

  • \begin{align}x^2+6x+8&=x^2+6x+9-9+8\\&=(x+3)^2-1\\&=(x+3+1)(x+3-1)\\&=(x+4)(x+2)\end{align}

  • \begin{align}x^2+8x+15&=x^2+8x+16-16+15\\&=(x+4)^2-1\\&=(x+4+1)(x+4-1)\\&=(x+5)(x+3)\end{align}

  • \begin{align}x^2+10x+24&=x^2+10x+25-25+24\\&=(x+5)^2-1\\&=(x+5+1)(x+5-1)\\&=(x+6)(x+4)\end{align}

  • \begin{align}
    x^2+12x+35&=x^2+12x+36-36+35\\
    &=(x+6)^2-1\\
    &=(x+6+1)(x+6-1)\\
    &=(x+7)(x+5)\end{align}

Oppgave 2

  • \begin{align}
    x^2+14x+48&=x^2+2\cdot 7\cdot x+7^2-7^2+48\\
    &=x^2+14x+49-49+48\\
    &=(x+7)^2-1\\
    &=(x+7-1)(x+7+1)\\
    &=(x+6)(x+8)
    \end{align}

  • \begin{align}
    x^2+16x+63
    &=x^2+2\cdot 8\cdot x+8^3-8^2+63\\
    &=(x+8)^2-1\\
    &=(x+8+1)(x+8-1)\\
    &=(x+9)(x+7)
    \end{align}

    $x^2+16x+63=(x+9)(x+7)$

  • \begin{align}
    x^2+10x+16
    &=x^2+2\cdot 5\cdot x+5^2-5^2+16\\
    &=(x+5)^2-3^2\\
    &=(x+5+3)(x+5-3)\\
    &=(x+8)(x+2)
    \end{align}

    $x^2+10x+16=(x+8)(x+2)$

  • \begin{align}
    x^2+12x+27
    &=x^2+2\cdot 6\cdot x+6^2-6^2+27\\
    &=(x+6)^2-36+27\\
    &=(x+6)^2-9\\
    &=(x+6)^2-3^2\\
    &=(x+6-3)(x+6+3)\\
    \end{align}

    $x^2+12x+27=(x+3)(x+9)$

  • \begin{align}
    x^2+12x+20
    &=x^2+3\cdot 6\cdot x+6^2-6^2+20\\
    &=(x+6)^2-36+20\\
    &=(x+6)^2-16\\
    &=(x+6)^2-4^2\\
    &=(x+6-4)(x+6+4)\\
    &=(x+2)(x+10)
    \end{align}

    $x^2+12x+20=(x+2)(x+10)$

  • \begin{align}
    x^2-2x-3&=(x-3)(x+1)
    \end{align}

  • \begin{align}x^2-2x-8=(x-4)(x+2)\end{align}

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

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

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

Oppgave 3

  • \begin{align}x^2-2x-48&=x^2-2x+1-1-48\\&=(x-1)^2-49\\&=(x-1+7)(x-1-7)\\&=(x+6)(x-8)\end{align}

  • \begin{align}x^2-2x-63&=x^2-2x+1-1-63\\&=(x-1)^2-64\\&=(x-1+8)(x-1-8)\\&=(x+7)(x-9)\end{align}

  • \begin{align}x^2+6x-16&=x^2+6x+9-9-16\\&=(x+3)^2-25\\&=(x+3+5)(x+3-5)\\&=(x+8)(x-2)\end{align}

  • \begin{align}x^2+6x-27&=x^2+6x+9-9-27\\&=(x+3)^2-36\\&=(x+3+6)(x+3-6)\\&=(x+9)(x-3)\end{align}

  • \begin{align}x^2+8x-20&=x^2+8x+16-16-20\\&=(x+4)^2-36\\&=(x+4+6)(x+4-6)\\&=(x+10)(x-2)\end{align}

Oppgave 4

  • \begin{align}x^2-4x+3=(x-3)(x-1)\end{align}

  • \begin{align}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^1-14x+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