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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

\[ \,\,\, 1) \quad x^2 +9x +18 \] \[ \,\,\, 2) \quad x^2 +11x +28 \] \[ \,\,\, 3) \quad x^2 +10x +25 \] \[ \,\,\, 4) \quad x^2 +17x +72 \] \[ \,\,\, 5) \quad x^2 +17x +72 \]

\[ \,\,\, 1) \quad x^2 +9x +18 = (x +6)(x +3) \]\[ \,\,\, 2) \quad x^2 +11x +28 = (x +4)(x +7) \]\begin{align} \,\,\, 3) \quad x^2 +10x +25 & = (x +5)(x +5) \\ & = (x +5)^2 \end{align}\[ \,\,\, 4) \quad x^2 +17x +72 = (x +8)(x +9) \]\[ \,\,\, 5) \quad x^2 +17x +72 = (x +9)(x +8) \]

Exercise #2 Hint

\[ \,\,\, 1) \quad x^2 -8x +12 \] \[ \,\,\, 2) \quad x^2 -9x +14 \] \[ \,\,\, 3) \quad x^2 -13x +42 \] \[ \,\,\, 4) \quad x^2 -7x +12 \] \[ \,\,\, 5) \quad x^2 -11x +24 \]

\[ \,\,\, 1) \quad x^2 -8x +12 = (x -2)(x -6) \]\[ \,\,\, 2) \quad x^2 -9x +14 = (x -2)(x -7) \]\[ \,\,\, 3) \quad x^2 -13x +42 = (x -6)(x -7) \]\[ \,\,\, 4) \quad x^2 -7x +12 = (x -4)(x -3) \]\[ \,\,\, 5) \quad x^2 -11x +24 = (x -3)(x -8) \]

Exercise #3 Hint

\[ \,\,\, 1) \quad x^2 -3x -10 \] \[ \,\,\, 2) \quad x^2 +x -42 \] \[ \,\,\, 3) \quad x^2 -4x -60 \] \[ \,\,\, 4) \quad x^2 +x -90 \] \[ \,\,\, 5) \quad x^2 +5x -36 \]

\[ \,\,\, 1) \quad x^2 -3x -10 = (x +2)(x -5) \]\[ \,\,\, 2) \quad x^2 +x -42 = (x +7)(x -6) \]\[ \,\,\, 3) \quad x^2 -4x -60 = (x +6)(x -10) \]\[ \,\,\, 4) \quad x^2 +x -90 = (x +10)(x -9) \]\[ \,\,\, 5) \quad x^2 +5x -36 = (x +9)(x -4) \]

Exercise #4

\[ \,\,\, 1) \quad x^2 -2x -24 \] \[ \,\,\, 2) \quad x^2 -13x +42 \] \[ \,\,\, 3) \quad x^2 -15x +50 \] \[ \,\,\, 4) \quad x^2 +12x +32 \] \[ \,\,\, 5) \quad x^2 +11x +10 \]

\[ \,\,\, 1) \quad x^2 -2x -24 = (x +4)(x -6) \]\[ \,\,\, 2) \quad x^2 -13x +42 = (x -6)(x -7) \]\[ \,\,\, 3) \quad x^2 -15x +50 = (x -5)(x -10) \]\[ \,\,\, 4) \quad x^2 +12x +32 = (x +8)(x +4) \]\[ \,\,\, 5) \quad x^2 +11x +10 = (x +10)(x +1) \]