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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

\[ \,\,\, 1) \quad x^2 +13x +42 \] \[ \,\,\, 2) \quad x^2 +5x +4 \] \[ \,\,\, 3) \quad x^2 +20x +100 \] \[ \,\,\, 4) \quad x^2 +18x +81 \] \[ \,\,\, 5) \quad x^2 +11x +10 \]

\[ \,\,\, 1) \quad x^2 +13x +42 = (x +6)(x +7) \]\[ \,\,\, 2) \quad x^2 +5x +4 = (x +1)(x +4) \]\begin{align} \,\,\, 3) \quad x^2 +20x +100 & = (x +10)(x +10) \\ & = (x +10)^2 \end{align}\begin{align} \,\,\, 4) \quad x^2 +18x +81 & = (x +9)(x +9) \\ & = (x +9)^2 \end{align}\[ \,\,\, 5) \quad x^2 +11x +10 = (x +10)(x +1) \]

Exercise #2 Hint

\[ \,\,\, 1) \quad x^2 -11x +28 \] \[ \,\,\, 2) \quad x^2 -10x +9 \] \[ \,\,\, 3) \quad x^2 -9x +18 \] \[ \,\,\, 4) \quad x^2 -11x +18 \] \[ \,\,\, 5) \quad x^2 -15x +54 \]

\[ \,\,\, 1) \quad x^2 -11x +28 = (x -7)(x -4) \]\[ \,\,\, 2) \quad x^2 -10x +9 = (x -9)(x -1) \]\[ \,\,\, 3) \quad x^2 -9x +18 = (x -3)(x -6) \]\[ \,\,\, 4) \quad x^2 -11x +18 = (x -2)(x -9) \]\[ \,\,\, 5) \quad x^2 -15x +54 = (x -9)(x -6) \]

Exercise #3 Hint

\[ \,\,\, 1) \quad x^2 +x -30 \] \[ \,\,\, 2) \quad x^2 +2x -8 \] \[ \,\,\, 3) \quad x^2 -2x -15 \] \[ \,\,\, 4) \quad x^2 -3x -40 \] \[ \,\,\, 5) \quad x^2 +3x -4 \]

\[ \,\,\, 1) \quad x^2 +x -30 = (x +6)(x -5) \]\[ \,\,\, 2) \quad x^2 +2x -8 = (x +4)(x -2) \]\[ \,\,\, 3) \quad x^2 -2x -15 = (x +3)(x -5) \]\[ \,\,\, 4) \quad x^2 -3x -40 = (x +5)(x -8) \]\[ \,\,\, 5) \quad x^2 +3x -4 = (x +4)(x -1) \]

Exercise #4

\[ \,\,\, 1) \quad x^2 -2x -48 \] \[ \,\,\, 2) \quad x^2 -14x +45 \] \[ \,\,\, 3) \quad x^2 -4x -60 \] \[ \,\,\, 4) \quad x^2 +18x +80 \] \[ \,\,\, 5) \quad x^2 +x -30 \]

\[ \,\,\, 1) \quad x^2 -2x -48 = (x +6)(x -8) \]\[ \,\,\, 2) \quad x^2 -14x +45 = (x -5)(x -9) \]\[ \,\,\, 3) \quad x^2 -4x -60 = (x +6)(x -10) \]\[ \,\,\, 4) \quad x^2 +18x +80 = (x +10)(x +8) \]\[ \,\,\, 5) \quad x^2 +x -30 = (x +6)(x -5) \]