(a minus sqrtx)^2=25
I have no idea of how to work this problem where i in up with the answers of a=7,-3. Basiclly I just gave it a guess. Could someone show the steps I need to take. Thanks.
take the sqrt of each side.
a-sqrtx=+- 5
now work each possible sign
sqrtx=a-5 or sqrt x=a+5
then square each side:
x=(a-5)^2 or x=(a+5)^2
Now you are presented with a quandry, resolve this way. If the domain of x is restricted to real numbers, then you cant take a sqrt of a negative numer, so
x>= 0 which means
(a-5)>0 or a>5 and (a+5)>0 a>-5 but the first predominates, so a>5 is the restriction on a.
what happen the the results of a=7, -3 to this problem because when I plug them in it equal to 25 in the end. Could you explain this a little more for me. Thanks.
To solve the equation,
(a - √x)^2 = 25
we can start by expanding the square term on the left side of the equation:
a^2 - 2a√x + (√x)^2 = 25
Simplifying this expression further:
a^2 - 2a√x + x = 25
Rearranging the equation:
x - 25 = 2a√x - a^2
Now, let's isolate the term involving the square root (√x) by moving the other terms to the right side of the equation:
2a√x = a^2 + 25 - x
To eliminate the square root, we need to square both sides of the equation:
(2a√x)^2 = (a^2 + 25 - x)^2
Simplifying the left side:
4a^2x = (a^2 + 25 - x)^2
Expanding the right side:
4a^2x = a^4 + 50a^2 + x^2 + 2a^2(25 - x) - 2ax(a^2 + 25 - x)
Rearranging this equation:
4a^2x - a^4 - 50a^2 - x^2 - 50a^2 + 2ax(a^2 + 25) - 2ax^2 - 50a^2 + 2ax^2 = 0
Combining like terms:
- a^4 - 2ax^2 + 4a^2x - 100a^2 + 2ax(a^2 + 25) - x^2 = 0
Now, let's group the like terms:
(- a^4 - 100a^2) + (2ax(a^2 + 25) - 2ax^2) + (4a^2x - x^2) = 0
Factoring out common terms:
-a^2(a^2 + 100) + 2ax(a^2 + 25) + x(4a^2 - x) = 0
Now, we have a quadratic equation in terms of a:
(a^2 + 100)a^2 - 2ax(a^2 + 25) + x(4a^2 - x) = 0
To solve for the values of a that satisfy this equation, we can either factor or use the quadratic formula. Factoring becomes complicated in this case, so we'll use the quadratic formula:
a = [-(-2ax(a^2 + 25)) ± √((-2ax(a^2 + 25))^2 - 4(a^2 + 100)(x(4a^2 - x)))] / (2(a^2 + 100))
Simplifying this equation further is quite involved and lengthy, but it can be done using algebraic manipulations. Alternatively, you can use numerical methods or a graphing calculator to approximate the values of a that satisfy the equation.
Please note that the above steps provide an algebraic explanation of solving the equation. In practice, if you're looking for specific numerical values of a, it is often easier to use a calculator or a computer program to find the solutions.