Certainly, let's solve the equation step-by-step.
We are given the quadratic equation:
[tex]\[ x^2 = 36 \][/tex]
Our aim is to find the value of [tex]\( x \)[/tex] that satisfies this equation. We do this by performing the same operation on both sides to maintain the equality, and in this case, the operation is taking the square root since we have [tex]\( x \)[/tex] raised to the second power (squared). When we take the square root of both sides, we obtain two solutions because the square root of a positive number has two possible values: one positive and one negative.
Let's solve it:
[tex]\[ x^2 = 36 \][/tex]
Take the square root of both sides.
[tex]\[ \sqrt{x^2} = \pm\sqrt{36} \][/tex]
The square root of [tex]\( x^2 \)[/tex] is [tex]\( x \)[/tex] (as [tex]\( x \)[/tex] can be either positive or negative, hence we include the ± symbol on the right side). The square root of 36 is 6. So we have:
[tex]\[ x = \pm6 \][/tex]
This means there are two solutions to the equation [tex]\( x^2 = 36 \)[/tex], which are:
[tex]\[ x = 6 \][/tex]
and
[tex]\[ x = -6 \][/tex]
So our solutions are [tex]\( x = 6 \)[/tex] and [tex]\( x = -6 \)[/tex]. Both of these values for [tex]\( x \)[/tex] will satisfy the original equation [tex]\( x^2 = 36 \)[/tex] because:
[tex]\[ (6)^2 = 36 \][/tex]
and
[tex]\[ (-6)^2 = 36 \][/tex]
Therefore, the two values of [tex]\( x \)[/tex] that solve the equation [tex]\( x^2 = 36 \)[/tex] are [tex]\( x = 6 \)[/tex] and [tex]\( x = -6 \)[/tex].
