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Quadratics


Monic quadratics
The general form of a quadratic is \( ax^2 + bx + c \). When the coefficient \(a = 1\), the quadratic is then referred to as monic.
To factorise a monic quadratic, find two numbers \(m\) and \(n\) such that \(m + n = b \) and \(m \times n = c \). Therefore: \( x^2 + bx + c = (x + m)(x + n) \)

Factorise the following expressions:

Exercise #1 Hint

\[ \,\,\, 1) \quad x^2 +5x +6 \] \[ \,\,\, 2) \quad x^2 +12x +36 \] \[ \,\,\, 3) \quad x^2 +10x +25 \] \[ \,\,\, 4) \quad x^2 +17x +70 \] \[ \,\,\, 5) \quad x^2 +5x +6 \]

\[ \,\,\, 1) \quad x^2 +5x +6 = (x +2)(x +3) \]\begin{align} \,\,\, 2) \quad x^2 +12x +36 & = (x +6)(x +6) \\ & = (x +6)^2 \end{align}\begin{align} \,\,\, 3) \quad x^2 +10x +25 & = (x +5)(x +5) \\ & = (x +5)^2 \end{align}\[ \,\,\, 4) \quad x^2 +17x +70 = (x +10)(x +7) \]\[ \,\,\, 5) \quad x^2 +5x +6 = (x +2)(x +3) \]

Exercise #2 Hint

\[ \,\,\, 1) \quad x^2 -14x +49 \] \[ \,\,\, 2) \quad x^2 -13x +30 \] \[ \,\,\, 3) \quad x^2 -6x +5 \] \[ \,\,\, 4) \quad x^2 -4x +3 \] \[ \,\,\, 5) \quad x^2 -11x +18 \]

\begin{align} \,\,\, 1) \quad x^2 -14x +49 & = (x -7)(x -7) \\ & = (x -7)^2 \end{align}\[ \,\,\, 2) \quad x^2 -13x +30 = (x -10)(x -3) \]\[ \,\,\, 3) \quad x^2 -6x +5 = (x -1)(x -5) \]\[ \,\,\, 4) \quad x^2 -4x +3 = (x -3)(x -1) \]\[ \,\,\, 5) \quad x^2 -11x +18 = (x -9)(x -2) \]

Exercise #3 Hint

\[ \,\,\, 1) \quad x^2 -4x -60 \] \[ \,\,\, 2) \quad x^2 -7x -30 \] \[ \,\,\, 3) \quad x^2 +5x -6 \] \[ \,\,\, 4) \quad x^2 -x -42 \] \[ \,\,\, 5) \quad x^2 +2x -48 \]

\[ \,\,\, 1) \quad x^2 -4x -60 = (x +6)(x -10) \]\[ \,\,\, 2) \quad x^2 -7x -30 = (x +3)(x -10) \]\[ \,\,\, 3) \quad x^2 +5x -6 = (x +6)(x -1) \]\[ \,\,\, 4) \quad x^2 -x -42 = (x +6)(x -7) \]\[ \,\,\, 5) \quad x^2 +2x -48 = (x +8)(x -6) \]

Exercise #4

\[ \,\,\, 1) \quad x^2 -3x -4 \] \[ \,\,\, 2) \quad x^2 -3x -70 \] \[ \,\,\, 3) \quad x^2 -5x -14 \] \[ \,\,\, 4) \quad x^2 +8x +7 \] \[ \,\,\, 5) \quad x^2 +9x +20 \]

\[ \,\,\, 1) \quad x^2 -3x -4 = (x +1)(x -4) \]\[ \,\,\, 2) \quad x^2 -3x -70 = (x +7)(x -10) \]\[ \,\,\, 3) \quad x^2 -5x -14 = (x +2)(x -7) \]\[ \,\,\, 4) \quad x^2 +8x +7 = (x +7)(x +1) \]\[ \,\,\, 5) \quad x^2 +9x +20 = (x +5)(x +4) \]