Find the solution set of 16𝑧 squared equals 32𝑧 minus 15 in the real numbers.
So the first thing that we should do is take everything to one side of the equation, so let’s move everything to the left.
So now we have 16𝑧 squared minus 32𝑧 plus 15 equals zero. Now we can solve this using the quadratic formula. So for a quadratic equation, that’s 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐, you solve by taking negative 𝑏 plus or minus the square root of 𝑏 squared minus four 𝑎𝑐 divided by two 𝑎.
So here we can see that 𝑎 is 16, 𝑏 is negative 32, and 𝑐 is 15. So plugging in 𝑎𝑏 and 𝑐, we get 𝑥 equals negative one times negative 32 plus or minus the square root of negative 32 squared minus four times 16 times 15, add that little parenthesis there, divided by two times 16. The two negatives cancel in the beginning, negative 32 squared is 1024, and negative four times 16 times 15 is negative 960. And then on the denominator, two times 16 is 32.
1024 minus 960 is equal to 64, and the square root of 64 is eight. So we have 32 plus eight divided by 32 and 32 minus eight divided by 32, which is 40 over 32 and 24 over 32, which reduces to five-fourths and three-fourths.