The square of a number is 12 less than 7 times the number. What is the number?

The number can either be 3 or 4. To solve this problem, write an equation that matches the description of the number, using a variable to represent the number, and solve for the variable. For example, let's use x as our variable. "The square of a number" means we need to square our variable, or multiply it by itself, which gives us x^2. We know this is equivalent to "12 less than 7 times the number", which means we need to multiply the variable by 7, then subtract 12 from it, which gives us 7x - 12. Since they are equivalent, we can say that x^2 = 7x - 12. To solve this equation, we want to set it equal to zero. We can accomplish this by subtracting 7x - 12 from both sides, giving us x^2 - 7x + 12 = 0. Now that the equation is in this form, we can reverse foil it by factoring, which gives us (x - 3) (x - 4) = 0. If we divide both sides by (x - 3), we get x - 4 = 0, thus x = 4, and if we divide both sides by (x - 4), we get x - 3 = 0, thus x = 3. To check these answers, we can plug them back into the original equation. If x = 4, then 4^2 = 7 (4) - 12, which simplifies to 16 = 28 - 12, and 16 = 16. Since both sides of the equation match, 4 is a valid answer. If x = 3, then 3^2 = 7 (3) - 12, which simplifies to 9 = 21 - 12, and 9 = 9. Both 3 and 4 are valid answers, so the number can either be 3 or 4.