Solve the equation using square roots. Select the solution(s).
[tex]4 x^2-351=20[/tex]
A. [tex]x=-15[/tex]
B. [tex]x=-16[/tex]
C. [tex]x=-14[/tex]
D. [tex]x=-8[/tex]
E. [tex]x=-6[/tex]
F. [tex]x=-1[/tex]
G. [tex]x=2[/tex]
H. [tex]x=4[/tex]
I. [tex]x=0[/tex]
J. [tex]x=12[/tex]
K. [tex]x=14[/tex]
L. [tex]x =16[/tex]
M. [tex]x=-12[/tex]
N. [tex]x=-2[/tex]
O. [tex]x=8[/tex]
P. [tex]x=18[/tex]
Q. [tex]x=-10[/tex]
R. [tex]x=0[/tex]
S. [tex]x=10[/tex]
T. no real solution
Answer (1)
Isolate the x 2 term: Add 351 to both sides and divide by 4, resulting in x 2 = 4 371 .
Take the square root of both sides: x = ± 4 371 .
Simplify the result: x = ± 2 371 ≈ ± 9.63 .
Since none of the given options match the calculated solutions, there are no real solutions from the list. no real solution
Explanation
Understanding the Problem We are given the equation 4 x 2 − 351 = 20 and asked to solve for x using square roots. We need to isolate x 2 and then take the square root of both sides.
Isolating the Term with x^2 First, we add 351 to both sides of the equation: 4 x 2 − 351 + 351 = 20 + 351
Simplifying the Equation This simplifies to: 4 x 2 = 371
Dividing by the Coefficient of x^2 Next, we divide both sides by 4: 4 4 x 2 = 4 371
Isolating x^2 This gives us: x 2 = 4 371
Taking the Square Root Now, we take the square root of both sides: x = ± 4 371
Simplifying the Square Root We can simplify this as: x = ± 4 371 = ± 2 371
Approximating the Value of x We can approximate the value of 371 as approximately 19.26. Therefore, x ≈ ± 2 19.26 ≈ ± 9.63
Checking the Options and Re-evaluating Looking at the given options, the closest values to ± 9.63 are x = − 10 and x = 10 . However, these are not in the provided table. The closest values in the table are x = − 8 and x = 8 . However, let's check if x = − 10 and x = 10 are indeed the solutions. If x = 10 , then 4 ( 10 ) 2 − 351 = 4 ( 100 ) − 351 = 400 − 351 = 49 = 20 .
If x = − 10 , then 4 ( − 10 ) 2 − 351 = 4 ( 100 ) − 351 = 400 − 351 = 49 = 20 .
Let's check the values x = − 8 and x = 8 .
If x = 8 , then 4 ( 8 ) 2 − 351 = 4 ( 64 ) − 351 = 256 − 351 = − 95 = 20 .
If x = − 8 , then 4 ( − 8 ) 2 − 351 = 4 ( 64 ) − 351 = 256 − 351 = − 95 = 20 .
Since none of the provided options are close to the actual solutions, we should re-evaluate the problem. We have x = ± 2 371 . We are looking for values of x in the table that satisfy the equation. Let's check x = 10 and x = − 10 again. 4 x 2 − 351 = 20 , so 4 x 2 = 371 , and x 2 = 4 371 = 92.75 . Then x = ± 92.75 ≈ ± 9.63 . From the table, the closest values are x = − 10 and x = 10 . However, we made a mistake in our calculations. We need to find the exact solutions. The exact solutions are x = ± 4 371 = ± 2 371 . Since 371 ≈ 19.26 , x ≈ ± 9.63 . None of the given options are correct.
Final Answer Since none of the provided options are close to the actual solutions, we can conclude that there are no real solutions from the given list.
Examples
When designing a square garden, you might need to determine the side length required to achieve a specific area. This involves solving an equation where the area is related to the square of the side length. For instance, if you want the garden's area to be 100 square feet, you would solve x 2 = 100 to find the side length x . Understanding how to solve such equations helps in practical applications like gardening, construction, and other fields where area and dimensions are important.
