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 $(document).ready(function(){$('#ex1').click(function(){ $('#hint1').toggle(500); }); }); Both $$m$$ and $$n$$ are positive $\,\,\, 1) \quad x^2 +10x +9$ $\,\,\, 2) \quad x^2 +15x +50$ $\,\,\, 3) \quad x^2 +9x +18$ $\,\,\, 4) \quad x^2 +11x +10$ $\,\,\, 5) \quad x^2 +19x +90$ $\,\,\, 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$(document).ready(function(){ $('#ex2').click(function(){$('#hint2').toggle(500); }); }); Both $$m$$ and $$n$$ are negative

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

$\,\,\, 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 $(document).ready(function(){$('#ex3').click(function(){ \$('#hint3').toggle(500); }); }); $$m$$ is postive and $$n$$ is negative

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

$\,\,\, 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)$