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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

\[ \,\,\, 1) \quad x^2 +16x +63 \] \[ \,\,\, 2) \quad x^2 +8x +16 \] \[ \,\,\, 3) \quad x^2 +13x +40 \] \[ \,\,\, 4) \quad x^2 +14x +49 \] \[ \,\,\, 5) \quad x^2 +10x +9 \]

\[ \,\,\, 1) \quad x^2 +16x +63 = (x +7)(x +9) \]\begin{align} \,\,\, 2) \quad x^2 +8x +16 & = (x +4)(x +4) \\ & = (x +4)^2 \end{align}\[ \,\,\, 3) \quad x^2 +13x +40 = (x +8)(x +5) \]\begin{align} \,\,\, 4) \quad x^2 +14x +49 & = (x +7)(x +7) \\ & = (x +7)^2 \end{align}\[ \,\,\, 5) \quad x^2 +10x +9 = (x +1)(x +9) \]

Exercise #2 Hint

\[ \,\,\, 1) \quad x^2 -8x +16 \] \[ \,\,\, 2) \quad x^2 -7x +12 \] \[ \,\,\, 3) \quad x^2 -7x +12 \] \[ \,\,\, 4) \quad x^2 -13x +36 \] \[ \,\,\, 5) \quad x^2 -8x +12 \]

\begin{align} \,\,\, 1) \quad x^2 -8x +16 & = (x -4)(x -4) \\ & = (x -4)^2 \end{align}\[ \,\,\, 2) \quad x^2 -7x +12 = (x -4)(x -3) \]\[ \,\,\, 3) \quad x^2 -7x +12 = (x -3)(x -4) \]\[ \,\,\, 4) \quad x^2 -13x +36 = (x -4)(x -9) \]\[ \,\,\, 5) \quad x^2 -8x +12 = (x -6)(x -2) \]

Exercise #3 Hint

\[ \,\,\, 1) \quad x^2 +x -90 \] \[ \,\,\, 2) \quad x^2 -6x -27 \] \[ \,\,\, 3) \quad x^2 -3x -70 \] \[ \,\,\, 4) \quad x^2 -4x -5 \] \[ \,\,\, 5) \quad x^2 -4x -60 \]

\[ \,\,\, 1) \quad x^2 +x -90 = (x +10)(x -9) \]\[ \,\,\, 2) \quad x^2 -6x -27 = (x +3)(x -9) \]\[ \,\,\, 3) \quad x^2 -3x -70 = (x +7)(x -10) \]\[ \,\,\, 4) \quad x^2 -4x -5 = (x +1)(x -5) \]\[ \,\,\, 5) \quad x^2 -4x -60 = (x +6)(x -10) \]

Exercise #4

\[ \,\,\, 1) \quad x^2 +3x -18 \] \[ \,\,\, 2) \quad x^2 -4x -45 \] \[ \,\,\, 3) \quad x^2 -2x -15 \] \[ \,\,\, 4) \quad x^2 +2x -63 \] \[ \,\,\, 5) \quad x^2 +x -6 \]

\[ \,\,\, 1) \quad x^2 +3x -18 = (x +6)(x -3) \]\[ \,\,\, 2) \quad x^2 -4x -45 = (x +5)(x -9) \]\[ \,\,\, 3) \quad x^2 -2x -15 = (x +3)(x -5) \]\[ \,\,\, 4) \quad x^2 +2x -63 = (x +9)(x -7) \]\[ \,\,\, 5) \quad x^2 +x -6 = (x +3)(x -2) \]