Answer :
Alright, let’s solve the given equation step-by-step by completing the square.
Given equation:
[tex]\[ -3x^2 - 298 = -60x - 1 \][/tex]
First, we need to move all terms to one side of the equation to set it equal to zero:
[tex]\[ -3x^2 + 60x - 297 = 0 \][/tex]
Next, normalize the equation by dividing all terms by [tex]\(-3\)[/tex], the coefficient of [tex]\(x^2\)[/tex]:
[tex]\[ x^2 - 20x + 99 = 0 \][/tex]
To complete the square, take the coefficient of [tex]\(x\)[/tex] (which is [tex]\(-20\)[/tex]), halve it, and then square it to find the term to complete the square:
[tex]\[ \left(\frac{-20}{2}\right)^2 = 100 \][/tex]
Add and subtract this number inside the equation:
[tex]\[ x^2 - 20x + 100 - 100 + 99 = 0 \implies (x - 10)^2 = 199 - 1 \implies (x - 10)^2 = 199 \][/tex]
So the intermediate step in the form [tex]\((x + a)^2 = b\)[/tex] is
[tex]\[ (x - 10.0)^2 = 199.33333333333331 \][/tex]
Therefore, the correct intermediate step is:
[tex]\[ (x - 10.0)^2 = 199.33333333333331 \][/tex]
Given equation:
[tex]\[ -3x^2 - 298 = -60x - 1 \][/tex]
First, we need to move all terms to one side of the equation to set it equal to zero:
[tex]\[ -3x^2 + 60x - 297 = 0 \][/tex]
Next, normalize the equation by dividing all terms by [tex]\(-3\)[/tex], the coefficient of [tex]\(x^2\)[/tex]:
[tex]\[ x^2 - 20x + 99 = 0 \][/tex]
To complete the square, take the coefficient of [tex]\(x\)[/tex] (which is [tex]\(-20\)[/tex]), halve it, and then square it to find the term to complete the square:
[tex]\[ \left(\frac{-20}{2}\right)^2 = 100 \][/tex]
Add and subtract this number inside the equation:
[tex]\[ x^2 - 20x + 100 - 100 + 99 = 0 \implies (x - 10)^2 = 199 - 1 \implies (x - 10)^2 = 199 \][/tex]
So the intermediate step in the form [tex]\((x + a)^2 = b\)[/tex] is
[tex]\[ (x - 10.0)^2 = 199.33333333333331 \][/tex]
Therefore, the correct intermediate step is:
[tex]\[ (x - 10.0)^2 = 199.33333333333331 \][/tex]
