Video Transcript
Find the solution set of the equation 𝑥 squared is equal to negative 18𝑥 plus six, giving values to three decimal places.
In order to solve this quadratic equation, we will use the quadratic formula. This states that if 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero, where 𝑎, 𝑏, and 𝑐 are constants and 𝑎 is nonzero, then 𝑥 is equal to negative 𝑏 plus or minus the square root of 𝑏 squared minus four 𝑎𝑐 all divided by two 𝑎. We note that for the equation to have real solutions 𝑏 squared minus four 𝑎𝑐, known as the discriminant, must be greater than or equal to zero.
In this question, our first step will be to rearrange the equation so it is in the correct form. We do this by adding 18𝑥 and subtracting six from both sides of the equation, such that 𝑥 squared plus 18𝑥 minus six equals zero. The values of 𝑎, 𝑏, and 𝑐 are one, 18, and negative six, respectively. They are the coefficients of 𝑥 squared, 𝑥, and then the constant term.
Substituting these values into the quadratic formula, 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. This simplifies to 𝑥 is equal to negative 18 plus or minus the square root of 348 all divided by two. Since 348 is equal to four multiplied by 87, we can rewrite the square root of 348 as the square root of four multiplied by the square root of 87. And as the square root of four is two, this, in turn, can be rewritten as two root 87. Next, we can divide the numerator and denominator of the right-hand side by two, giving us negative nine plus or minus the square root of 87.
There are two solutions to our quadratic equation: 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. These are equal to 0.327379 and so on and negative 18.327379 and so on.
We are asked to give our answers to three decimal places. The solution set of the equation 𝑥 squared is equal to negative 18𝑥 plus six contains the values 0.327 and negative 18.327 correct to three decimal places.