Let's solve the given equation step-by-step through the process of completing the square.
The given equation is:
[tex]\[ 5x^2 + 380 = 90x - 10 \][/tex]
First, move all terms to one side of the equation so that we set it to zero:
[tex]\[ 5x^2 + 380 - 90x + 10 = 0 \][/tex]
[tex]\[ 5x^2 - 90x + 390 = 0 \][/tex]
Next, we will complete the square. Begin by making the coefficient of [tex]\(x^2\)[/tex] equal to 1 by factoring out the 5:
[tex]\[ 5(x^2 - 18x + 78) = 0 \][/tex]
Now, take the coefficient of [tex]\(x\)[/tex] (which is -18), halve it, and square the result:
[tex]\[ \text{Half of } -18 \text{ is } -9 \][/tex]
[tex]\[ (-9)^2 = 81 \][/tex]
Add and subtract this square term inside the parenthesis:
[tex]\[ 5(x^2 - 18x + 81 - 81 + 78) = 0 \][/tex]
which simplifies to:
[tex]\[ 5((x - 9)^2 - 3) = 0 \][/tex]
Distribute the 5 back:
[tex]\[ 5((x - 9)^2 - 3) = 0 \][/tex]
[tex]\[ 5 \cdot (x - 9)^2 - 15 = 0 \][/tex]
[tex]\[ 5 \cdot (x-9)^2 = 15 \][/tex]
Finally, divide both sides by 5:
[tex]\[ (x-9)^2 = 3 \][/tex]
So, the intermediate step in the form [tex]\((x + a)^2 = b\)[/tex] is:
[tex]\[ \boxed{(x-9)^2 = 3} \][/tex]
