Let's solve the quadratic equation 5n squared plus 19n plus 18 equals 0. First, we identify the coefficients: a equals 5, b equals 19, and c equals 18. We'll use the quadratic formula to find the solutions.
Now we calculate the discriminant using the formula delta equals b squared minus 4ac. Substituting our values: delta equals 19 squared minus 4 times 5 times 18, which gives us 361 minus 360, equals 1. Since the discriminant is positive, we have two distinct real solutions.
Step 3: Now we apply the quadratic formula. n equals negative b plus or minus the square root of delta, all divided by 2a. Substituting our values: n equals negative 19 plus or minus the square root of 1, divided by 2 times 5. This simplifies to n equals negative 19 plus or minus 1, all divided by 10.
Step 4: Now we calculate both solutions. For the first solution, we use plus 1: n1 equals negative 19 plus 1, divided by 10, which equals negative 18 over 10, or negative 9 fifths. For the second solution, we use minus 1: n2 equals negative 19 minus 1, divided by 10, which equals negative 20 over 10, or negative 2. Therefore, our solutions are n equals negative 2 and n equals negative 9 fifths.
Let's verify our solutions by substituting them back into the original equation. For n equals negative 2: 5 times negative 2 squared plus 19 times negative 2 plus 18 equals 20 minus 38 plus 18, which equals 0. For n equals negative 9 fifths, the calculation also equals 0. Both solutions are verified. Therefore, the final answer is n equals negative 2 and n equals negative 9 fifths.