give exact and approximate solutions to three decimal places x^2 - 7x + 7 = 0
To find the exact and approximate solutions to the equation x^2 - 7x + 7 = 0, we can use the quadratic formula.
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 1, b = -7, and c = 7. Substituting these values into the quadratic formula:
x = (-(-7) ± √((-7)^2 - 4(1)(7))) / (2(1))
x = (7 ± √(49 - 28)) / 2
x = (7 ± √21) / 2
Now we have two options, one for each of the ± signs:
1. First, let's consider the positive sign:
x = (7 + √21) / 2
Calculating this expression, we get:
x ≈ 6.791
2. Now, let's consider the negative sign:
x = (7 - √21) / 2
Calculating this expression, we get:
x ≈ 0.209
Therefore, the exact solutions to the equation x^2 - 7x + 7 = 0 are:
x = (7 + √21) / 2
x = (7 - √21) / 2
And the approximate solutions, rounded to three decimal places, are:
x ≈ 6.791
x ≈ 0.209