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Solving quadratic equations using the quadratic formula
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Which of these equations could you solve using the quadratic equation formula?
Hi Kim! Are you lying there immersed in your pile of formulae? Found a smart formula? Oh, yes! With that one, you can solve quadratic equations!
It looks a bit weird, but with a little explanation you can soon grasp it. Then with it, you can solve every quadratic equation in existence! Shall we jump right in? Here's the formula again. Now we'll look at the equation: 'three x-squared' plus 'five-x' plus 'two' equals 'zero' Let's put them together to see how it all works.
Compare the equation and formula. In front of 'x-squared' in the formula, we have the coefficient 'a'... and in the equation there is a 'three' in front of 'x-squared'... ... so we know that 'a' equals 'three'. And we make a note of that.
And 'b' before the 'x' term in the formula... Yes, that must be the 'five' in the equation. We know 'b' equals 'five' And 'c' is the constant term. That represents 'two' in the equation. So that's known, so we can make a note of that too. 'c' equals 'two'. 'A' and 'b' are the coefficients in front of the x-terms. 'c' is the constant term.
We know the quantities of 'a', 'b', and 'c'. And 'x' is the unknown, which we are calculating. The next step is to put these values into the formula. That's what it means to apply a formula. In place of the letters 'a', 'b' and 'c', we put in the values we know.
Here we have the formula. And here are the solutions. 'x' equals... Minus 'b'... and 'b' equals 'five'... as we wrote down...
plus-or-minus the square-root of... 'square-root of 'b', in other words 'square-root of five' ... minus four times 'a' times 'c' 'a' is 'three' ... and 'c' is 'two'. All this is divided by 'two' times 'a' -- In this case, 'two' times 'three'. So we see that 'x' equals 'minus five' Plus-or-minus the square-root of 'twenty-five minus twenty-four' divided by 'six'.
The first solution: 'x-one', is equal to 'minus five plus one' divided by 'six' equals 'minus four' divided by 'six' and we can reduce that to: 'minus two-thirds'. The second solution: 'x-two', is equal to 'minus five minus one' divided by 'six'... 'x-two' equals 'minus one'. And there we have two solutions, two results: 'x' equals 'minus two-thirds' and 'x' equals 'minus one'. You can pause the film here and check the solutions are correct. We'll look at one more example: What is the solution of...
minus 'four x-squared' plus 'six x' equals 'two'? If we look at the formula we see that all the terms are gathered on the left side, and the right side is zero. For the formula to be useful, the equation must follow the same pattern. We gather all the terms on the left-hand-side, like this... and compare with the formula: In this equation, 'a' equals 'minus four' 'b' equals 'six' ...
and 'c' equals 'minus two' ... and now we go to the formula: 'x' equals -- long division sign -- And plus-or-minus and a long square-root sign And because 'a' is in two places, we put in 'minus four' there and there. 'b' equals 'six', and that also goes in two places. There, with a minus sign in front and there. And 'c': 'minus two' goes in there. And we remember the 'minus-four' from the formula.
That's goes there. And the 'two'. That goes there. Here we have the equation's solutions. Now we do a little calculation, then we're finished.
It's vital to keep all the minus signs organised. 'Minus' times 'minus' is 'plus'. Remember that. 'x' equals 'minus six' plus-or-minus the root of 'thirty-six minus thirty-two' divided by 'minus eight' That equals 'minus six' plus-or-minus the root of 'four' divided by 'minus eight' 'X-1' is therefore 'minus six' plus 'two' divided by 'minus eight'... Equals 'minus four' divided by 'minus eight'. 'Minus' divided by 'minus'... that's 'plus'. And plus 'four eighths':... 'a half'. 'X-2' is Minus 'six' minus 'two' Divided by minus 'eight' Equals 'minus eight' Divided by 'minus eight'.
Again: minus divided by minus... 'X-two' is plus... One! So with the help of this formula you can solve every quadratic equation in existence. Gather all the terms on the left hand side. And remember: keep track of the plus and minus signs!