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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

\[ \,\,\, 1) \quad x^2 +10x +9 \] \[ \,\,\, 2) \quad x^2 +15x +50 \] \[ \,\,\, 3) \quad x^2 +9x +18 \] \[ \,\,\, 4) \quad x^2 +11x +10 \] \[ \,\,\, 5) \quad x^2 +19x +90 \]

\[ \,\,\, 1) \quad x^2 +10x +9 = (x +9)(x +1) \]\[ \,\,\, 2) \quad x^2 +15x +50 = (x +10)(x +5) \]\[ \,\,\, 3) \quad x^2 +9x +18 = (x +6)(x +3) \]\[ \,\,\, 4) \quad x^2 +11x +10 = (x +1)(x +10) \]\[ \,\,\, 5) \quad x^2 +19x +90 = (x +9)(x +10) \]

Exercise #2 Hint

\[ \,\,\, 1) \quad x^2 -13x +40 \] \[ \,\,\, 2) \quad x^2 -10x +24 \] \[ \,\,\, 3) \quad x^2 -11x +18 \] \[ \,\,\, 4) \quad x^2 -9x +8 \] \[ \,\,\, 5) \quad x^2 -10x +25 \]

\[ \,\,\, 1) \quad x^2 -13x +40 = (x -8)(x -5) \]\[ \,\,\, 2) \quad x^2 -10x +24 = (x -6)(x -4) \]\[ \,\,\, 3) \quad x^2 -11x +18 = (x -2)(x -9) \]\[ \,\,\, 4) \quad x^2 -9x +8 = (x -1)(x -8) \]\begin{align} \,\,\, 5) \quad x^2 -10x +25 & = (x -5)(x -5) \\ & = (x -5)^2 \end{align}

Exercise #3 Hint

\[ \,\,\, 1) \quad x^2 +2x -3 \] \[ \,\,\, 2) \quad x^2 +6x -7 \] \[ \,\,\, 3) \quad x^2 -3x -70 \] \[ \,\,\, 4) \quad x^2 -5x -14 \] \[ \,\,\, 5) \quad x^2 +5x -36 \]

\[ \,\,\, 1) \quad x^2 +2x -3 = (x +3)(x -1) \]\[ \,\,\, 2) \quad x^2 +6x -7 = (x +7)(x -1) \]\[ \,\,\, 3) \quad x^2 -3x -70 = (x +7)(x -10) \]\[ \,\,\, 4) \quad x^2 -5x -14 = (x +2)(x -7) \]\[ \,\,\, 5) \quad x^2 +5x -36 = (x +9)(x -4) \]

Exercise #4

\[ \,\,\, 1) \quad x^2 +8x +15 \] \[ \,\,\, 2) \quad x^2 -8x -9 \] \[ \,\,\, 3) \quad x^2 -14x +45 \] \[ \,\,\, 4) \quad x^2 -4x -45 \] \[ \,\,\, 5) \quad x^2 +13x +36 \]

\[ \,\,\, 1) \quad x^2 +8x +15 = (x +5)(x +3) \]\[ \,\,\, 2) \quad x^2 -8x -9 = (x +1)(x -9) \]\[ \,\,\, 3) \quad x^2 -14x +45 = (x -5)(x -9) \]\[ \,\,\, 4) \quad x^2 -4x -45 = (x +5)(x -9) \]\[ \,\,\, 5) \quad x^2 +13x +36 = (x +9)(x +4) \]