answer:To find the minimum possible value of ( left(x^{2}+6x+2right)^{2} ) over all real numbers (x), we proceed with the following steps: 1. **Rewrite the expression inside the square:** First, consider the quadratic expression (x^2 + 6x + 2). We can complete the square for this quadratic expression to simplify it. [ x^2 + 6x + 2 = left( x^2 + 6x + 9 right) - 7 = left( x + 3 right)^2 - 7 ] 2. **Substitute the completed square form back into the original expression:** [ left( x^2 + 6x + 2 right)^2 = left( left( x + 3 right)^2 - 7 right)^2 ] 3. **Analyze the expression ( left( left( x + 3 right)^2 - 7 right)^2 ):** The square of any real number is always non-negative. Therefore, the expression inside the outer square, ( left( x + 3 right)^2 - 7 ), should ideally be zero or as close to zero as possible to minimize the value. 4. **Find the point where the expression is zero:** For ( left( x + 3 right)^2 - 7 = 0 ): [ left( x + 3 right)^2 = 7 ] Solving this equation gives: [ x + 3 = pm sqrt{7} ] Thus, the values of (x) that make the expression zero are: [ x = -3 + sqrt{7} quad text{and} quad x = -3 - sqrt{7} ] 5. **Substitute these values back to find the minimum possible value:** [ left( left( -3 + sqrt{7} + 3 right)^2 - 7 right)^2 = left( left( sqrt{7} right)^2 - 7 right)^2 = left( 7 - 7 right)^2 = 0^2 = 0 ] Similarly, for (x = -3 - sqrt{7}): [ left( left( -3 - sqrt{7} + 3 right)^2 - 7 right)^2 = left( left( -sqrt{7} right)^2 - 7 right)^2 = left( 7 - 7 right)^2 = 0^2 = 0 ] Thus, the minimum possible value of ( left( x^2 + 6x + 2 right)^2 ) is (0), and this minimum is achieved when ( x = -3 + sqrt{7} ) or ( x = -3 - sqrt{7} ). # Conclusion: The minimum possible value is [ boxed{0} ]

answer:To solve this problem, let's break down the purchases and the money spent step by step: 1. **Total Money Spent on Glasses:** Peter starts with 50 and is left with 1, which means he spent 50 - 1 = 49 on glasses. 2. **Money Spent on Small Glasses:** Each small glass costs 3, and he buys 8 of them. Therefore, the total cost for small glasses is 3 times 8 = 24. 3. **Money Spent on Large Glasses:** After buying small glasses, the remaining amount spent on large glasses is the total spent on glasses minus the amount spent on small glasses. This gives us 49 - 24 = 25 spent on large glasses. 4. **Number of Large Glasses Bought:** Each large glass costs 5. To find out how many large glasses he bought, we divide the total spent on large glasses by the cost of one large glass: 25 / 5 = 5. Therefore, Peter bought boxed{5} large glasses.

answer:Let's denote the original price of the shirt as P. After the first discount of 33.333%, the price of the shirt becomes: P - (33.333% of P) = P - (0.33333 * P) = 0.66667 * P After the second discount of 25% on the already discounted price, the price becomes: 0.66667 * P - (25% of 0.66667 * P) = 0.66667 * P - (0.25 * 0.66667 * P) = 0.75 * 0.66667 * P = 0.5 * P After the third discount of 20% on the double discounted price, the price becomes: 0.5 * P - (20% of 0.5 * P) = 0.5 * P - (0.2 * 0.5 * P) = 0.8 * 0.5 * P = 0.4 * P We know that the final price after all the discounts is 15, so we can set up the equation: 0.4 * P = 15 To find the original price P, we divide both sides of the equation by 0.4: P = 15 / 0.4 P = 37.50 Therefore, the original price of the shirt before any discounts was boxed{37.50} .

answer:Jen's son is currently 16 years old. Three times her son's present age would be: 3 * 16 = 48 years old. Jen is currently 41 years old. To find out how many years less Jen's present age is than 3 times her son's present age, we subtract her age from the tripled age of her son: 48 - 41 = 7 years. Jen's present age is boxed{7} years less than 3 times her son's present age.