Solve x2=78 by inspection. There are two real solutions. Enter the lesser number first, and use exponents in your responses.
To solve the equation x^2 = 78, we can take the square root of both sides to find the values of x. However, before proceeding, we need to note that there may be two real solutions for this equation.
Taking the square root of both sides, we have:
√(x^2) = √(78)
Since x^2 is positive for all real numbers, we can ignore the negative square root.
x = √(78)
Using a calculator, we can determine that √(78) is approximately 8.83176.
Thus, the lesser of the two real solutions to the equation x^2 = 78 is approximately 8.83176.
To solve for x in the equation x^2 = 78 by inspection, we need to find the two real numbers whose square is 78.
Looking at the perfect squares, we can see that 9^2 = 81 and 8^2 = 64. Since 78 is between these two perfect squares, the two real solutions will be between 8 and 9.
Let's calculate the square root of 78 by estimating:
√78 ≈ √64 = 8
Therefore, the first real solution is 8. Now, we can find the other solution by taking the negative value (-x) of the first solution:
-x = -8
So, the second real solution is -8.
Therefore, the two real solutions are -8 and 8.
To solve the equation x^2 = 78 by inspection, we need to find two real solutions.
To do this, let's consider perfect squares that are close to 78. We know that 9^2 = 81 and 8^2 = 64. Since 78 is between these two numbers, we can estimate the solutions to be between 8 and 9.
Now, we can narrow down the possibilities by trying different numbers within this range and checking if their squares are close to 78.
Let's start with 8. If we square 8, we get 64, which is less than 78. Increasing the value of x will result in a larger square, so we need to try a number greater than 8.
Next, let's try 8.5. Squaring 8.5 gives us 72.25, which is still less than 78. Our squared result is getting closer, so we need to try a number slightly larger.
Finally, let's try 8.7. Squaring 8.7 gives us 75.69, which is also less than 78. We need to try an even larger number.
Based on this analysis, we can conclude that the value of x that satisfies x^2 = 78 lies between 8.7 and 8.8.
Since we want the lesser number first, we can round down to 8.7 as the first solution.
To find the second solution, we can take the negative value of the first solution, so the second solution is -8.7.
Therefore, the solutions to the equation x^2 = 78, listed in ascending order, are -8.7 and 8.7.