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Question Video: Determining the Real Solutions of Quadratic Equations Mathematics • 9th Grade

Solve 3𝑥² + 2𝑥 + 4 = 0.

02:12

Video Transcript

Solve three 𝑥 squared plus two 𝑥 plus four is equal to zero. So if we just write out our quadratic formula there, we can see that 𝑎 is three, 𝑏 is two, and 𝑐 is four. So plugging those values for 𝑎, 𝑏, and 𝑐 into the quadratic formula gives us 𝑥 is equal to negative two plus or minus the square root of two squared minus four times three times four all over two times three. Well, two squared is four, four times three times four is forty-eight, so we’ve got four take away forty-eight in that square root. And then two times three on the denominator is six. So two possible solutions, negative two plus the square root of negative forty-four all over six, or 𝑥 is equal to negative two minus the square root of negative forty-four all over six. But if you try typing that into your calculator, you’ll get a math error. The problem is this bit here, square root of negative forty-four. There aren’t any real numbers that you can multiply by themselves to give a negative answer. If you take a negative number and multiply it by itself, you get a positive answer. And if you take a positive number and you multiply it by itself, you get a positive answer. So there isn’t a real number that you can multiply by itself that will give a negative answer.

Now if we have a look at the graph of 𝑦 equals three 𝑥 squared plus two 𝑥 plus four, and then we put 𝑦 equal to zero. We can see that, in fact, there aren’t any 𝑥-values that are gonna generate a 𝑦-coordinate of zero. That curve doesn’t cut through the 𝑥-axis here. There are no 𝑥-values that generate a 𝑦-coordinate of zero.

So this was a bit of a trick question that broke the formula. We were asked to solve something that doesn’t have any real solutions. So it seems like a bit of a trick question, but really it’s just saying, you know, where does this quadratic curve cut the 𝑥-axis on a curve that doesn’t cut the 𝑥-axis. So that’s what we found out. So when you’ve got a negative value here for 𝑏 squared minus four 𝑎𝑐, you know you’ve got a curve that doesn’t cut the 𝑥-axis.

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