WEBVTT
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Find, in its simplest form, the quadratic equation whose roots are negative three and negative eight.
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As it tells us that it’s a quadratic equation in the question, we actually know that we’re actually — if we were gonna factor it, we’d have a pair of parentheses.
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And also, as we know the roots of the equation, in that case, we can actually say that they’d be equal to zero.
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What we need to do now is actually to find out first what is inside our parentheses.
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And we know that for the equation to equal zero, one of the parentheses must also equal zero.
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So we can use that to actually find out what’s gonna go inside the parentheses.
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And we can do that because we know that the roots, so the 𝑥-values, are either negative three or negative eight.
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So first of all, we can use negative three.
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And we know that negative three plus something is equal to zero.
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And we know that that’ll be negative three plus positive three.
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So therefore, we can actually include our first parentheses.
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So therefore, our first parentheses is 𝑥 plus three.
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Now we just need to find out what the second one could be.
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And now we need to see what will add to negative eight to give us zero, because that’s the other one of our roots.
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Well, the answer is positive eight.
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So therefore, great.
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We’ve now found what’s gonna go into our second parentheses, and that’s 𝑥 plus eight because we have positive eight from solving “what would add to negative eight to make zero”.
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So great!
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We’ve now got the fact that, fully factored, our quadratic would be 𝑥 plus three, 𝑥 plus eight.
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And we can double check that because we can see that negative three plus three would be zero.
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That would make it totally equal to zero, the whole equation.
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And negative eight plus eight would also make the equation equal to zero.
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Fantastic!
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Now we’d want to actually expand our parentheses.
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To do that, first of all, we’re gonna multiply the 𝑥 terms.
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So that’s 𝑥 multiplied by 𝑥 which gives us 𝑥 squared.
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And then I multiply 𝑥 by positive eight which gives us plus eight 𝑥.
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And then I multiply positive three by 𝑥 which gives us plus three 𝑥.
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And then finally, positive three multiplied by positive eight which gives us plus 24.
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Fantastic!
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We’ve now expanded our parentheses and we’re actually almost there to the final answer.
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But if we look at the question, it says find, in its simplest form, the quadratic equation.
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So now what we need to do is collect like terms.
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So all we can see is that’ll be equal to 𝑥 squared and then plus.
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And we’ve got two like terms here.
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We’ve got eight 𝑥 and three 𝑥.
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So positive eight 𝑥 plus three 𝑥 which can give us positive 11𝑥 or plus 11𝑥.
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And then, just plus 24.
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So therefore, we can say that, in its simplest form, the quadratic equation whose roots are negative three and negative eight is going to be equal to 𝑥 squared plus 11𝑥 plus 24.