How do you solve using the completing the square method $z^{2} + 10 z = - 24$ $z^2 + 10z = -24$ ?

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Answer 1 · 2016-03-04T13:56:55

The solutions are:

$z = - 4$ $color(green)(z = -4$
$z = - 6$ $color(green)(z =-6$

$z^{2} + 10 z = - 24$ $z^2+10z= -24$

To write the Left Hand Side as a Perfect Square, we add 25 to both sides
$z^{2} + 10 z + 25 = - 24 + 25$ $z^2+10z +25= -24 +25$

$z^{2} + 2 \cdot 5 \cdot z + 5^{2} = 1$ $z^2 + 2*5*z + 5^2 = 1$

Using the Identity ${(a + b)}^{2} = a^{2} + 2 a b + b^{2}$ $color(blue)((a+b)^2 = a^2 + 2ab + b^2$ , we get
${(z + 5)}^{2} = 1$ $(z+5)^2 = 1$

$z + 5 = + \sqrt{1} = 1$ $z + 5 = + sqrt1 =1$
$z = 1 - 5 = - 4$ $color(green)(z =1 - 5= -4$

or,

$z + 5 = - \sqrt{1} = - 1$ $z +5 = -sqrt1= -1$
$z = - 1 - 5 = - 6$ $color(green)(z = -1 -5=-6$

How do you solve using the completing the square method z2+10z=−24z^2 + 10z = -24?