Factorising quadratic equations

Some quadratic equations can be factorised completely. I’ll explain with an example.

$x^2 + 5x + 6 = 0$

This equation can be solved with the method of squares that we have used as follows but with a complication because half of 5 is 2.5.

$(x + 2.5)^2 = x^2 + 5x + 2.5^2$

$(x + 2.5)^2 = x^2 + 5x + 6.25$

$x^2 + 5x = (x + 2.5)^2 - 6.25$

Then by substitution

$x^2 + 5x + 6 = 0$

$(x + 2.5)^2 - 6.25 + 6 = 0$

$(x + 2.5)^2 - 0.25 = 0$

$(x + 2.5)^2 = 0.25$

$sqrt{(x + 2.5)^2} = sqrt{0.25}$

$x + 2.5 = pm0.5$

$x + 2.5 = -2.5pm0.5$

$x = -2, x = -3$

This could have been done in an easier way by factorising completely as follows

$x^2 + 5x + 6 = 0$

$(x + 2)(x + 3) = 0$

As a check, we can expand these brackets to get

$x(x + 3) + 2(x + 3) = 0$

$x^2+ 3x + 2x + 6 = 0$

$x^2+ 5x + 6 = 0$

so we know we have factorised the equation correctly

We know that $(x + 2)(x + 3) = 0$ when either $(x + 2) = 0$ or $(x + 3) = 0$ . This is because when we multiply any number by zero we get zero and that zero is the only number we can multiply a non-zero number by to get zero.

$x + 2 = 0$ means $x = -2$

$x + 3 = 0$ means $x = -3$

$x^2 + 5x + 6 = 0$ when $x = -2$ and $x = -3$

We can see this on the plot of $y = x^2 + 5x + 6$ below. When $y = 0$ , $x^2 + 5x + 6 = 0$ . We can see on the plot that this happens at $x = -2$ and $x = -3$ .

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