How do I work this problem, it is not a homework problem but a problem similar to a homework problem. I know i need to factor the left side of the problem. here is what I have so far..
x^2-2x+1=7
x^2-2x+1-7=7-7
x^2-2x-6=0
here is where I run into an issue, I can not seem to fatctor it down.
There are times where they do not factor nicely. A quick check is to calculate the sqrt(b^2 - (4*a*c)) part of the quadratic equation. Here it looks like that would be sqrt(28) or 2*sqrt(7). This pretty much says it is not easily factored by inspection. Use the complete quadratic formula for the answer.
To factor the expression x^2 - 2x - 6 = 0, you can use the quadratic formula or try to factor it by finding two numbers whose product is the constant term (-6) and whose sum is the coefficient of the middle term (-2).
In this case, the constant term is -6 and the coefficient of the middle term is -2. We are looking for two numbers that multiply to -6 and add up to -2.
One approach is to list all the factor pairs of -6 and then find the pair that adds up to -2:
1 * (-6) = -6
2 * (-3) = -6
From these factor pairs, we can see that the pair (2, -3) adds up to -1, not -2. Therefore, it seems that this quadratic equation does not factor nicely using integer factors.
If you still want to solve the equation, you can use the quadratic formula, which states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
For the equation x^2 - 2x - 6 = 0, the coefficients are a = 1, b = -2, and c = -6. Substituting these into the quadratic formula, we get:
x = (-(-2) ± √((-2)^2 - 4(1)(-6))) / (2(1))
x = (2 ± √(4 + 24)) / 2
x = (2 ± √28) / 2
Simplifying further, we have:
x = (2 ± 2√7) / 2
x = 1 ± √7
Therefore, the solutions to the equation x^2 - 2x - 6 = 0 are x = 1 + √7 and x = 1 - √7.