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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

\[ \,\,\, 1) \quad x^2 +16x +64 \] \[ \,\,\, 2) \quad x^2 +16x +64 \] \[ \,\,\, 3) \quad x^2 +10x +9 \] \[ \,\,\, 4) \quad x^2 +15x +50 \] \[ \,\,\, 5) \quad x^2 +11x +30 \]

\begin{align} \,\,\, 1) \quad x^2 +16x +64 & = (x +8)(x +8) \\ & = (x +8)^2 \end{align}\begin{align} \,\,\, 2) \quad x^2 +16x +64 & = (x +8)(x +8) \\ & = (x +8)^2 \end{align}\[ \,\,\, 3) \quad x^2 +10x +9 = (x +9)(x +1) \]\[ \,\,\, 4) \quad x^2 +15x +50 = (x +5)(x +10) \]\[ \,\,\, 5) \quad x^2 +11x +30 = (x +6)(x +5) \]

Exercise #2 Hint

\[ \,\,\, 1) \quad x^2 -11x +18 \] \[ \,\,\, 2) \quad x^2 -13x +42 \] \[ \,\,\, 3) \quad x^2 -10x +21 \] \[ \,\,\, 4) \quad x^2 -12x +20 \] \[ \,\,\, 5) \quad x^2 -10x +16 \]

\[ \,\,\, 1) \quad x^2 -11x +18 = (x -2)(x -9) \]\[ \,\,\, 2) \quad x^2 -13x +42 = (x -6)(x -7) \]\[ \,\,\, 3) \quad x^2 -10x +21 = (x -7)(x -3) \]\[ \,\,\, 4) \quad x^2 -12x +20 = (x -2)(x -10) \]\[ \,\,\, 5) \quad x^2 -10x +16 = (x -2)(x -8) \]

Exercise #3 Hint

\[ \,\,\, 1) \quad x^2 -8x -20 \] \[ \,\,\, 2) \quad x^2 +3x -70 \] \[ \,\,\, 3) \quad x^2 -x -12 \] \[ \,\,\, 4) \quad x^2 +2x -15 \] \[ \,\,\, 5) \quad x^2 -x -42 \]

\[ \,\,\, 1) \quad x^2 -8x -20 = (x +2)(x -10) \]\[ \,\,\, 2) \quad x^2 +3x -70 = (x +10)(x -7) \]\[ \,\,\, 3) \quad x^2 -x -12 = (x +3)(x -4) \]\[ \,\,\, 4) \quad x^2 +2x -15 = (x +5)(x -3) \]\[ \,\,\, 5) \quad x^2 -x -42 = (x +6)(x -7) \]

Exercise #4

\[ \,\,\, 1) \quad x^2 +12x +32 \] \[ \,\,\, 2) \quad x^2 -5x -50 \] \[ \,\,\, 3) \quad x^2 +10x +9 \] \[ \,\,\, 4) \quad x^2 -15x +50 \] \[ \,\,\, 5) \quad x^2 -2x -3 \]

\[ \,\,\, 1) \quad x^2 +12x +32 = (x +8)(x +4) \]\[ \,\,\, 2) \quad x^2 -5x -50 = (x +5)(x -10) \]\[ \,\,\, 3) \quad x^2 +10x +9 = (x +9)(x +1) \]\[ \,\,\, 4) \quad x^2 -15x +50 = (x -5)(x -10) \]\[ \,\,\, 5) \quad x^2 -2x -3 = (x +1)(x -3) \]