What is the solutions to X2 equals 8
The Correct Answer and Explanation is:
To solve the equation ( x^2 = 8 ), we need to isolate ( x ). The equation is a simple quadratic equation where ( x ) is squared. Here are the steps to find the solutions:
- Isolate ( x^2 ): The equation is already in a suitable form, with ( x^2 ) on one side and ( 8 ) on the other.
- Take the square root of both sides: To solve for ( x ), we take the square root of both sides of the equation. However, it’s important to remember that taking the square root can yield both a positive and a negative solution. Thus, we write:
[
x = \pm \sqrt{8}
] - Simplify ( \sqrt{8} ): The square root of ( 8 ) can be simplified further. We know that:
[
\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}
]
Therefore, we can rewrite our solutions as:
[
x = \pm 2\sqrt{2}
] - Final solutions: This means that the solutions to the equation ( x^2 = 8 ) are:
[
x = 2\sqrt{2} \quad \text{and} \quad x = -2\sqrt{2}
]
Explanation:
Quadratic equations, like ( x^2 = c ) (where ( c ) is a positive number), will always yield two solutions due to the nature of squaring a number. When ( x ) is squared, both ( x ) and ( -x ) will produce the same positive value, ( c ). In this case, squaring ( 2\sqrt{2} ) or ( -2\sqrt{2} ) gives ( 8 ), confirming that both values are valid solutions.
Quadratic equations can be graphed as parabolas, which are U-shaped curves. The equation ( y = x^2 ) would intersect the horizontal line ( y = 8 ) at two points: ( x = 2\sqrt{2} ) and ( x = -2\sqrt{2} ). This graphical representation reinforces the idea that there are two solutions.
Thus, the complete solution set for the equation ( x^2 = 8 ) is:
[
x = 2\sqrt{2} \quad \text{and} \quad x = -2\sqrt{2}
]