solve the equation
(x-5)^2=10
x-5 = + sqrt 10 or - sqrt 10
x= 5 + sqrt 10
or
x = 5 - sqrt 10
To solve the equation (x - 5)^2 = 10, we need to find the value(s) of x that make the equation true. Here's how to do it step by step:
Step 1: Expand the equation
(x - 5)^2 = 10
x^2 - 10x + 25 = 10
Step 2: Subtract 10 from both sides
x^2 - 10x + 25 - 10 = 10 - 10
x^2 - 10x + 15 = 0
Step 3: Factor the quadratic equation
To factor the quadratic equation x^2 - 10x + 15 = 0, we need to find two numbers that multiply to give 15 and add up to give -10. The numbers -5 and -3 fit these criteria.
(x - 5)(x - 3) = 0
Step 4: Set each factor equal to zero
x - 5 = 0 or x - 3 = 0
Step 5: Solve for x in each equation
For x - 5 = 0, add 5 to both sides:
x = 5
For x - 3 = 0, add 3 to both sides:
x = 3
Step 6: Check your solutions
To confirm if the solutions are correct, substitute both values back into the original equation:
For x = 5:
(5 - 5)^2 = 10
0^2 = 10
0 = 10 (Not true)
For x = 3:
(3 - 5)^2 = 10
(-2)^2 = 10
4 = 10 (Not true)
Hence, there are no real solutions to the equation (x - 5)^2 = 10.