Final answer:
The equation 1 + 7z + z² = 0 can be solved using the quadratic formula to find two real solutions. The solutions are obtained by substituting the coefficients into the formula, calculating the discriminant, and simplifying the resulting expressions.
Step-by-step explanation:
To find all real solutions of the equation 1 + 7z + z² = 0, we can use the quadratic formula. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and x represents the variable. In this case, our equation is in terms of z, so we have a = 1, b = 7, and c = 1.
The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a). Plugging our values into the formula, we get:
z = (-7 ± √(7² - 4∗1∗1)) / (2∗1)
z = (-7 ± √(49 - 4)) / 2
z = (-7 ± √(45)) / 2
Since 45 is a positive number, we can calculate its square root, meaning we will have two real solutions. After simplifying, we find:
z = (-7 + √45) / 2 and z = (-7 - √45) / 2
By calculating these expressions, we obtain two real solutions, which can be expressed to an appropriate number of significant figures. We should also check the answer to see if it is reasonable.