Math

17. The Zero Product Property states that if a*b = 0, then either a=0 or b=0. In this equation, we have -8n(5n+3). We set the product of -8n and (5n+3) equal to 0 and then solve for two separate equations. Multiply the parentheses and distribute the -8n: -8n(5n+3) = 0 -8n^2 - 24n = 0 Set each factor equal to 0 and solve for n. -8n^2 = 0 --> n = 0 -24n = 0 --> n = 0 Therefore, n = 0. 19. To solve this equation by factoring, first bring all terms to one side and equate to zero: 5z^2 - 2z - 3 = 0 To factor, we look for two numbers that multiply together to make -3 and add to give -2. The numbers -3 and 1 fit the bill. So, factor the left side as: (5z + 3)(z - 1) = 0 We can set each factor equal to 0 and solve for z: 5z + 3 = 0 --> z = -3/5