Let's solve the equation step by step to find the values of [tex]\( x \)[/tex] that make the equation true.
The given equation is:
[tex]\[ 10x^2 - 64 = 36 + 6x^2 \][/tex]
First, let's simplify this equation by bringing all terms to one side of the equation:
[tex]\[ 10x^2 - 64 - 36 - 6x^2 = 0 \][/tex]
Combine like terms:
[tex]\[ (10x^2 - 6x^2) - 100 = 0 \][/tex]
[tex]\[ 4x^2 - 100 = 0 \][/tex]
Next, let's solve for [tex]\( x \)[/tex]:
[tex]\[ 4x^2 - 100 = 0 \][/tex]
Add 100 to both sides:
[tex]\[ 4x^2 = 100 \][/tex]
Divide both sides by 4:
[tex]\[ x^2 = 25 \][/tex]
Take the square root of both sides:
[tex]\[ x = \pm 5 \][/tex]
We have two solutions for this equation:
[tex]\[ x = 5 \][/tex]
[tex]\[ x = -5 \][/tex]
Now, let's check each given value to see if it satisfies the original equation:
A. [tex]\( x = -5 \)[/tex]
[tex]\[ 10(-5)^2 - 64 = 36 + 6(-5)^2 \][/tex]
[tex]\[ 250 - 64 = 36 + 150 \][/tex]
[tex]\[ 186 = 186 \][/tex]
This is true, so [tex]\( x = -5 \)[/tex] is a solution.
B. [tex]\( x = -\sqrt{5} \)[/tex]
[tex]\[ 10(-\sqrt{5})^2 - 64 = 36 + 6(-\sqrt{5})^2 \][/tex]
[tex]\[ 10(5) - 64 = 36 + 6(5) \][/tex]
[tex]\[ 50 - 64 = 36 + 30 \][/tex]
[tex]\[ -14 = 66 \][/tex]
This is false, so [tex]\( x = -\sqrt{5} \)[/tex] is not a solution.
C. [tex]\( x = 10 \)[/tex]
[tex]\[ 10(10)^2 - 64 = 36 + 6(10)^2 \][/tex]
[tex]\[ 1000 - 64 = 36 + 600 \][/tex]
[tex]\[ 936 \neq 636 \][/tex]
This is false, so [tex]\( x = 10 \)[/tex] is not a solution.
D. [tex]\( x = 5 \)[/tex]
[tex]\[ 10(5)^2 - 64 = 36 + 6(5)^2 \][/tex]
[tex]\[ 250 - 64 = 36 + 150 \][/tex]
[tex]\[ 186 = 186 \][/tex]
This is true, so [tex]\( x = 5 \)[/tex] is a solution.
E. [tex]\( x = -10 \)[/tex]
[tex]\[ 10(-10)^2 - 64 = 36 + 6(-10)^2 \][/tex]
[tex]\[ 1000 - 64 = 36 + 600 \][/tex]
[tex]\[ 936 \neq 636 \][/tex]
This is false, so [tex]\( x = -10 \)[/tex] is not a solution.
F. [tex]\( x = \sqrt{5} \)[/tex]
[tex]\[ 10(\sqrt{5})^2 - 64 = 36 + 6(\sqrt{5})^2 \][/tex]
[tex]\[ 10(5) - 64 = 36 + 6(5) \][/tex]
[tex]\[ 50 - 64 = 36 + 30 \][/tex]
[tex]\[ -14 = 66 \][/tex]
This is false, so [tex]\( x = \sqrt{5} \)[/tex] is not a solution.
Therefore, the values of [tex]\( x \)[/tex] that are solutions to the equation are:
[tex]\[ \boxed{-5, 5} \][/tex]
