91Ó°ÊÓ

Solve each equation. $$ (x-5)(x+7)=0 $$

Solve each equation. $$ (x+5)(x-7)=0 $$

Factor each quadratic expression that can be factored using integers. Identify those that cannot, and explain why they can't be factored. $$ 2 x^{2}-8 x-10 $$

In Exercises 22 and \(23,\) write an equation to represent the situation. Economics the balance \(b\) in a savings account at the end of any year \(t\) if \(\$ 5,000\) is deposited initially and the account earns 8\(\%\) interest per year

History When the famous German mathematician Gauss was a young boy, he amazed his teacher by rapidly computing the sum of the integers from 1 to 100. He realized that he could compute the sum without adding all the numbers, by grouping the 100 numbers into pairs. To see a shortcut for finding this sum, look at two lists of 1 to 100, one in reverse order. \(\begin{array}{cccccccccccc}{1} & {2} & {3} & {4} & {5} & {6} & {7} & {\dots} & {50} & {\dots} & {94} & {95} & {96} & {97} & {98} & {99} & {100} \\ {100} & {99} & {98} & {97} & {96} & {95} & {94} & {\dots} & {51} & {\dots} & {7} & {6} & {5} & {4} & {3} & {2} & {1}\end{array}\) a. What is the sum of each pair? b. How many pairs are there? c. What is the sum of all these pairs? d. How many times is each of the integers from 1 to 100 counted in this sum? e. Consider your answers to Parts c and d. What is the sum of the integers from 1 to 100? f. Explain how you can use this same reasoning to find the sum of the integers from 1 to n for any value of n. Write a formula for s, the sum of the first n positive integers. g. Chloe added several consecutive numbers, starting at 1, and found a sum of 91. Write an equation you could use to find the numbers she added. Solve your equation by completing the square. Check your answer with the formula.

Expand each expression. $$ (d+3)(d+6) $$

Identify the values of \(a, b,\) and \(c\) in each equation by rearranging it into the form of the general quadratic equation, \(a x^{2}+b x+c=0\) $$ 2 x^{2}-7 x=-5 $$

Expand each expression. $$ (3 k-2 m)^{2} $$

Challenge When you simplify algebraic expressions, sometimes the simplified expression is not equivalent to the original for all values of the variable. For example, consider this expression: $$\frac{5 a+10}{a^{2}-4}$$ a. Factor the denominator. For what values of \(a\) is the expression undefined? That is, for what values is the denominator equal to 0\(?\) b. Now write the expression above using factored forms for both the numerator and denominator. Be sure to look for common factors in the terms. c. Simplify the fraction. d. Now try to evaluate the fraction using each value that made the original expression undefined. You found those values in Part a.) e. You should have seen in Part d the simplified fraction is not equivalent to the original fraction for all values of a. Explain why this happened. f. When you simplify an algebraic fraction, you should note any values of the variable that make the simplified fraction unequal to the original. For example, the fraction \(\frac{x(x+1)}{3 x}\) can be simplified as \(\frac{x+1}{3},\) where \(x \neq 0\) . Simplify the fraction \(\frac{2 m+1}{4 m^{2}-1}\)

The Numkenas built a small, square patio from square bricks with sides of length 1 foot. They bought just enough bricks to build the patio, but after they built it they decided it was too small. To extend the length and width of the patio by \(d\) feet, they had to buy 24 more bricks. The original side length of the patio was 5 feet. a. Draw a diagram to represent this situation. Be sure to show both the original patio and the new one. b. Write an equation to represent this situation. c. Simplify your equation and solve it to find \(d\) , the amount by which the patio's length and width were increased. Check your answer.