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Factorising quadratic equations (completing the square)

Quadratic equations Factorising quadratic equations (completing the square)

We can use what we have learnt about factorising to solve quadratic equations.

Let’s take the equation $x^2 + 6x + 2 = 0$

We know that $(x + 3)^2 = x^2 + 6x + 9$

We can rearrange this to help us solve $x^2 + 6x + 2 = 0$

$x^2 + 6x + 9 = (x + 3)^2$

$x^2 + 6x = (x + 3)^2 - 9$

We can substitute $(x + 3)^2 - 9$ for $x^2 + 6x$ in $x^2 + 6x + 2 = 0$ like this

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

$(x + 3)^2 - 9 + 2 = 0$

$(x + 3)^2 - 7 = 0$

$(x + 3)^2 = 7$

$sqrt{(x + 3)^2} = pmsqrt{7}$

$x + 3 = pmsqrt{7}$

$x = -3pmsqrt{7}$

This method of solving quadratic equations is called Completing the Square. This is because by squaring a term like $x + 3$ , we create a square like in the picture below. It is the blue $3^2 = 9$ that completes the square. $(x + 3)^2 = x^2 + 6x + 9$ .

In the equation we solved above, $x^2 + 6x + 2 = 0$ , we have to remove this 9 when we substitute $(x + 3)^2$ into the equation. This is the step where we go from $x^2 + 6x + 2 = 0$ to $(x + 3)^2 - 9 + 2 = 0$ . We are, in fact, subtracting the part that “completes the square”.

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