Find the solution set of the
equation negative 𝑥 squared minus one over negative two 𝑥 minus six is equal to
one-fifth for all real values, giving values to one decimal place.
We will begin by simplifying our
equation. To do this, we will cross multiply
or multiply both sides of the equation by negative two 𝑥 minus six and by five. On the left-hand side, we have five
multiplied by negative 𝑥 squared minus one. And on the right-hand side, we have
one multiplied by negative two 𝑥 minus six. We can then distribute the
parentheses, also known as expanding the brackets. This gives us negative five 𝑥
squared minus five is equal to negative two 𝑥 minus six.
We will then add five 𝑥 squared
and five to both sides of this equation. This leaves us with the equation
zero is equal to five 𝑥 squared minus two 𝑥 minus one. As our equation is in the form 𝑎𝑥
squared plus 𝑏𝑥 plus 𝑐 is equal to zero, we can solve it using the quadratic
formula. This states that 𝑥 is equal to
negative 𝑏 plus or minus the square root of 𝑏 squared minus four 𝑎𝑐 all divided
by two 𝑎. The positive and negative signs
give us two solutions.
Our values of 𝑎, 𝑏, and 𝑐 are
five, negative two, and negative one, respectively. This means that 𝑥 is equal to
negative negative two plus or minus the square root of negative two squared minus
four multiplied by five multiplied by negative one all divided by two multiplied by
five. This simplifies to one plus or
minus the square root of six all divided by five.
We have two possible solutions: 𝑥
is equal to one plus root six over five and 𝑥 is equal to one minus root six over
five. Typing these into the calculator
gives us 𝑥 is equal to 0.6898 and so on and 𝑥 is equal to negative 0.2898 and so
on. We are asked to give our values to
one decimal place. This means that 𝑥 can be equal to
0.7 or negative 0.3. Writing these in set notation, we
see that the solution set of the equation negative 𝑥 squared minus one over
negative two 𝑥 minus six equals one-fifth is 0.7 and negative 0.3.