Question Video: Finding the Solution Set of a Quadratic Equation Mathematics

Find the solution set of the equation (−𝑥² − 1)/(−2𝑥 − 6) = 1/5 in ℝ, giving values to one decimal place.

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Video Transcript

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.

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