Final answer:
The equation 'Eight less than the square of a number is the same as adding the number and four' can be translated to the quadratic equation z^2 - 8 = z + 4, which simplifies to z^2 - z - 12 = 0. This equation can be solved by methods such as factoring, completing the square, or using the quadratic formula.
Step-by-step explanation:
To set up the equation for the statement 'Eight less than the square of a number is the same as adding the number and four', we need to translate this into a mathematical representation. Let the unknown number be represented by the variable z. The square of a number is z squared, or z2. Eight less than the square of a number is then represented as z2 - 8. According to the statement, this value is the same as adding the number (which is z) to four, written as z + 4. Therefore, the equation we need to solve is:
Equation:
z2 - 8 = z + 4
To solve this quadratic equation, you would first move all the terms to one side of the equation to set it to zero. So, subtract z and 4 from both sides to get:
z2 - z - 12 = 0
This is a standard form quadratic equation that can be solved by factoring, completing the square, or using the quadratic formula. However, we can often find the solution by factoring if it simplifies nicely. In this case, you would look for two numbers that multiply to -12 and add to -1 (the coefficient of z). Once you find these numbers, you can write the factors of the quadratic equation and solve for the value(s) of z.