Find the solution set of the equation 𝑥 squared is equal to negative 18𝑥 plus six, giving values to three decimal places.
We will begin by rearranging our equation so it is in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 is equal to zero, where 𝑎, 𝑏, and 𝑐 are constants and 𝑎 is not equal to zero. To do this, we will add 18𝑥 and subtract six from both sides. This gives us 𝑥 squared plus 18𝑥 minus six is equal to zero. The quadratic is now written in a form we can solve, where 𝑎 is equal to one, 𝑏 is equal to 18, and 𝑐 is negative six.
The quadratic cannot be factorized. Therefore, we will use the quadratic formula to solve it. This states that 𝑥 is equal to negative 𝑏 plus or minus the square root of 𝑏 squared minus four 𝑎𝑐 all over two 𝑎. The positive and negative signs in front of the square root enable us to get two solutions. Substituting in our values, we have 𝑥 is equal to negative 18 plus or minus the square root of 18 squared minus four multiplied by one multiplied by negative six all divided by two multiplied by one.
Using our calculator, we can see that this can be written in radical or surd form as 𝑥 is equal to negative nine plus or minus the square root of 87. This gives us two possible solutions, either 𝑥 is equal to negative nine plus the square root of 87 or 𝑥 is equal to negative nine minus the square root of 87. Negative nine plus the square root of 87 is equal to 0.3273 and so on. And negative nine minus the square root of 87 is equal to negative 18.3273 and so on.
We need to give our answers to three decimal places. Therefore, 𝑥 is equal to 0.327 or negative 18.327. We can write these answers using set notation as shown. The solution set of the equation 𝑥 squared is equal to negative 18𝑥 plus six is 0.327 and negative 18.327.