Find the solution set of the equation three 𝑥 squared minus two times seven minus 𝑥 equals zero, giving values to one decimal place if necessary.
As we look at this equation, we realize that it’s not in standard form. So that’s the first thing we’ll need to do. To give the equation in standard form, we’ll need to distribute this negative two to the seven and the negative 𝑥. Negative two times seven equals negative 14. We can also write minus 14; that will work. Then we’ll multiply negative two times negative 𝑥, and that gives us a positive two 𝑥. Bring down the equal to zero. But we still not fully in standard form.
Standard form has our variables decreasing by the degree. We would start with three 𝑥 squared plus two 𝑥 minus 14 equals zero. Because we have a leading coefficient that’s not one, it will probably be easiest to solve this equation using the quadratic formula. The quadratic formula says this: 𝑥 equals negative b plus or minus the square root of 𝑏 squared minus four 𝑎𝑐 over two 𝑎. This is true for equations in the format 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero.
In order to solve this problem, it’s only a matter of plugging in all the variables in their correct places. But to do that, we’ll need to identify our 𝑎, 𝑏, and 𝑐 values. In our case, 𝑎 equals three, 𝑏 equals two. Be careful with this one, 𝑐 doesn’t equal 14; it equals negative 14. The quadratic equation is written with the format wel- where 𝑏𝑥 is being added to 𝑐. In our case, we’re subtracting 𝑐. So what we can do is we can say negative 14.
Now it’s a matter of us carefully plugging in each piece of information. 𝑥 equals negative two plus or minus the square root of two squared minus four times three times negative 14 all over two times three. Let’s see if we can do some simplification here. Negative two plus or minus two squared, equals four, minus four times three times negative 14 which equals negative 168 over six. We can still reduce a little bit further before we enter it into our calculator. This time I wanna say four minus negative 168 which equals 172 all over six.
But our question is asking to give the values to one decimal place. This means we’ll actually need to solve this problem using a calculator. In our calculator, we’ll need to enter negative two plus the square root of 172 divided by six. We’ll also need to enter negative two minus the square root of 172 divided by six. Once we enter the top one into our calculator, we get 1.85247. Rounding that to one decimal place, we get 1.9.
Our second solution yields negative 2.519146 and on and on and on. Rounded to the nearest decimal place, is negative two and a half, negative 2.5.
Using the quadratic formula, we found the solution set of our equation to be: 𝑥 equals 1.9 and negative two and a half.