# Determine a from the equation a*a=-4 if a*b=ab+5(a+b)+21 .

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a*a = -4

a*b = ab + 5(a+b) + 21

Now let us substitutw with a:

a*a = a^2 + 5(a+a) + 21

= a^2 + 10a + 21 = -4

==> a^2 + 10a + 25 = 0

Now factor:

==> (a+5)^2 = 0

==> **a = -5**

To determine the unknown from the given equation, we need the definition of the operation of the composition law, definition which is given by the expression from enunciation.

We have to put the given expression in a more convenient manner.

a*b=(a+5)(b+5)-4

Now, we'll try to solve the equation a*a=-4

a*a=(a+5)(a+5)-4

-4=a^2+10a+25 - 4

a^2+10a+25=0

(a+5)^2=0

**a1=a2=-5**

To solve for a in the equation,a*a =-4 . The composition is a*b = ab+5(a+b)+21.

We rewrite the equation by using the composition a inplace of b.

a*a = a*a+5(a+a)+25 = -4

a^2+10a+25-4 = 0

a^2+10a+21 = 0(a^2+7a

a^2+7a +3a +21 = 0

a(a+7)+3(a+7) = 0. Factor out a+7

(a+7)(a+3) = 0.

a+7 =0 gives a =-7

a+3 =0 gives a =-3.

So a=-7 or -3