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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

\[ \,\,\, 1) \quad x^2 +13x +30 \] \[ \,\,\, 2) \quad x^2 +20x +100 \] \[ \,\,\, 3) \quad x^2 +6x +5 \] \[ \,\,\, 4) \quad x^2 +6x +5 \] \[ \,\,\, 5) \quad x^2 +13x +40 \]

\[ \,\,\, 1) \quad x^2 +13x +30 = (x +3)(x +10) \]\begin{align} \,\,\, 2) \quad x^2 +20x +100 & = (x +10)(x +10) \\ & = (x +10)^2 \end{align}\[ \,\,\, 3) \quad x^2 +6x +5 = (x +5)(x +1) \]\[ \,\,\, 4) \quad x^2 +6x +5 = (x +5)(x +1) \]\[ \,\,\, 5) \quad x^2 +13x +40 = (x +8)(x +5) \]

Exercise #2 Hint

\[ \,\,\, 1) \quad x^2 -11x +10 \] \[ \,\,\, 2) \quad x^2 -5x +4 \] \[ \,\,\, 3) \quad x^2 -12x +20 \] \[ \,\,\, 4) \quad x^2 -18x +80 \] \[ \,\,\, 5) \quad x^2 -11x +24 \]

\[ \,\,\, 1) \quad x^2 -11x +10 = (x -1)(x -10) \]\[ \,\,\, 2) \quad x^2 -5x +4 = (x -4)(x -1) \]\[ \,\,\, 3) \quad x^2 -12x +20 = (x -10)(x -2) \]\[ \,\,\, 4) \quad x^2 -18x +80 = (x -10)(x -8) \]\[ \,\,\, 5) \quad x^2 -11x +24 = (x -8)(x -3) \]

Exercise #3 Hint

\[ \,\,\, 1) \quad x^2 -4x -12 \] \[ \,\,\, 2) \quad x^2 -6x -27 \] \[ \,\,\, 3) \quad x^2 +8x -20 \] \[ \,\,\, 4) \quad x^2 -2x -15 \] \[ \,\,\, 5) \quad x^2 +4x -12 \]

\[ \,\,\, 1) \quad x^2 -4x -12 = (x +2)(x -6) \]\[ \,\,\, 2) \quad x^2 -6x -27 = (x +3)(x -9) \]\[ \,\,\, 3) \quad x^2 +8x -20 = (x +10)(x -2) \]\[ \,\,\, 4) \quad x^2 -2x -15 = (x +3)(x -5) \]\[ \,\,\, 5) \quad x^2 +4x -12 = (x +6)(x -2) \]

Exercise #4

\[ \,\,\, 1) \quad x^2 -13x +30 \] \[ \,\,\, 2) \quad x^2 -16x +63 \] \[ \,\,\, 3) \quad x^2 +3x -54 \] \[ \,\,\, 4) \quad x^2 -6x -40 \] \[ \,\,\, 5) \quad x^2 -2x +1 \]

\[ \,\,\, 1) \quad x^2 -13x +30 = (x -3)(x -10) \]\[ \,\,\, 2) \quad x^2 -16x +63 = (x -7)(x -9) \]\[ \,\,\, 3) \quad x^2 +3x -54 = (x +9)(x -6) \]\[ \,\,\, 4) \quad x^2 -6x -40 = (x +4)(x -10) \]\begin{align} \,\,\, 5) \quad x^2 -2x +1 & = (x -1)(x -1) \\ & = (x -1)^2 \end{align}