0:00:00.650,0:00:02.700
We're asked to solve[br]the quadratic equation,

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negative 3x squared plus[br]10x minus 3 is equal to 0.

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And it's already written[br]in standard form.

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And there's many[br]ways to solve this.

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But in particular, all solve[br]it using the quadratic formula.

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So let me just rewrite it.

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We have negative 3x squared[br]plus 10x minus 3 is equal to 0.

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And actually, I'll[br]solve it twice

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using the quadratic[br]formula to show you

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that as long as we manipulated[br]this in the valid way,

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the quadratic[br]formula will give us

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the exact same roots or[br]the exact same solutions

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to this equation.

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So in this form right over[br]here, what are our ABCs?

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Let's just remind ourselves[br]what the quadratic formula even

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is actually.

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That's a good place to start.

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The quadratic formula[br]tells us that if we

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have a quadratic[br]equation in the form ax

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squared plus bx plus c is equal[br]to 0, so in standard form,

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then the roots of this are[br]x are equal to negative b

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plus or minus the[br]square root of b

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squared minus 4ac,[br]all of that over 2a.

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And this is derived from[br]completing the square

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in a general way.

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So it's no magic here, and I've[br]derived it in other videos.

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But this is the[br]quadratic formula.

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This is actually giving[br]you two solutions,

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because you have the[br]positive square root here

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and the negative square root.

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So let's apply it here in the[br]case where-- in this case,

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a is equal to negative[br]3, b is equal to 10,

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and c is equal to negative 3.

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So applying the quadratic[br]formula right here,

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we get our solutions to be[br]x is equal to negative b.

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b is 10.

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So negative b is negative 10[br]plus or minus the square root

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of b squared.

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b is 10.

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So b squared is 100[br]minus 4 times a times c.

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So minus 4 times negative[br]3 times negative 3.

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Let me just write it down.

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Minus 4 times negative[br]3 times negative 3.

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All of that's under[br]the radical sign.

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And then all of that is over 2a.

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So 2 times a is negative 6.

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So this is going to be[br]equal to negative 10

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plus or minus the square root[br]of 100 minus-- negative 3 times

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negative 3 is positive 9.

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Positive 9 times[br]4 is positive 36.

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We have a minus sign out here.

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So minus 36.

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All of that over negative 6.

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This is equal to[br]100 minus 36 is 64.

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So negative 10 plus or[br]minus the square root of 64.

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All of that over negative 6.

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The principal square[br]root of 64 is 8.

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But we're taking the positive[br]and negative square root.

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So this is negative 10 plus[br]or minus 8 over negative 6.

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So if we take the[br]positive version,

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we say x could be[br]equal to-- negative 10

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plus 8 is negative[br]2 over negative 6.

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So that was taking[br]the plus version.

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That's this right over here.

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And negative 2 over[br]negative 6 is equal to 1/3.

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If we take the[br]negative square root,

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negative 10 minus 8-- So let's[br]take negative 10 minus 8.

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That would be x is equal[br]to-- negative 10 minus 8

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is negative 18.

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And that's going to[br]be over negative 6.

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Negative 18 divided by[br]negative 6 is positive 3.

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So the two roots for[br]this quadratic equation

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are positive 1/3 and positive 3.

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And I want to show you the[br]we'll get the same answer,

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even if we manipulate this.

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Some people might[br]not like the fact

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that our first coefficient[br]here is a negative 3.

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Maybe they want a positive 3.

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So to get rid of[br]that negative 3,

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they can multiply both sides of[br]this equation times negative 1.

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And then if you did[br]that, you would get 3x

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squared minus 10x plus 3 is[br]equal to 0 times negative 1,

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which is still equal to 0.

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So in this case, a is equal to[br]3, b is equal to negative 10,

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and c is equal to 3 again.

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And we could apply[br]the quadratic formula.

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We get x is equal to[br]negative b. b is negative 10.

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So negative negative[br]10 is positive

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10, plus or minus[br]the square root

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of b squared, which is[br]negative 10 squared,

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which is 100, minus[br]4 times a times c.

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a times c is 9 times 4 is 36.

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So minus 36.

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All of that over 2 times a.

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All of that over 6.

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So this is equal to 10 plus or[br]minus the square root of 64,

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or really that's[br]just going to be 8.

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All of that over 6.

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If we add 8 here, we get[br]10 plus 8 is 18 over 6.

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We get x could be equal to 3.

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Or if we take the negative[br]square root or the negative 8

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here, 10 minus 8 is 2.

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2 over 6 is 1/3.

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So once again, you get[br]the exact same solutions.