Algebra (all content)
- The quadratic formula
- Understanding the quadratic formula
- Using the quadratic formula
- Worked example: quadratic formula
- Worked example: quadratic formula (example 2)
- Worked example: quadratic formula (negative coefficients)
- Quadratic formula
- Using the quadratic formula: number of solutions
- Number of solutions of quadratic equations
- Proof of the quadratic formula
- Quadratic formula review
- Discriminant review
- Quadratic formula proof review
Worked example: quadratic formula (negative coefficients)
Sal solves -3x^2+10x-3=0 by plugging a=-3, b=10, c=-3 in the quadratic formula. Then he multiplies everything by -1 and solves again. The results are the same! Created by Sal Khan and Monterey Institute for Technology and Education.
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- What do I do if the number inside the square root is not a perfect square? For example: 10x^2 -9x -6 = 0(8 votes)
- You have to see if there are any perfect square factors of the number that can be taken out. So b^2 - 4ac = (-9)^2 - 4(10)(-6) = 81 + 240 = 321. I can divide by three to get 3*107, but there are no other factors, so all you can do is leave it as √321, so the answers will be x = (9 ± √321)/20.(10 votes)
- Where did he get the second equation from?(7 votes)
- how do you remember the formula? I keep forgetting.(5 votes)
- The more you say it, write it, and hear it, the quicker it will get into short term and long term memory, so keep doing all three of these things. I hate it when my students do not want to say things out loud because they think it is silly, but it works. Practice with trying to tell your mom or dad, they may remember it from when they were in high school.(5 votes)
- When we have a quadratic equation or expression, does it matter to which side we minus or subtract the terms to match ax squared + bx + c?
Do we have to pull the terms to the left side?(5 votes)
- What does Sal mean by "the roots of this equation?" I just think of it being the solutions of the equation.(3 votes)
- "Roots" of an equation means the values of x that satisfy the equation ax^2 + bx + c = 0. So yes it's more like the solution to the quadratic equation.(6 votes)
- is the quadtratic formula applicable to all quadratic equation?(3 votes)
- Yes, it works for all quadratic equations. You just need to simplify the quadratic and write it in the form: Ax^2 + Bx + C = 0 before using the formula.(4 votes)
- When you start with a negative b like in the equation -6x^2 -3x +6 and you are trying to insert it in the equation with the a, b, and c's do you keep -3 or change it to 3 for the first -b?
Sry if it doesn't make sense(3 votes)
- In the quadratic formula, it’s -b, so when you plug in, you would do -(-3) which is same thing as +3.(4 votes)
- What would you do if in the square root, there was a negative?(5 votes)
- Why did -2/6 become a positive 1/3? Wouldn't 1/3 be negative?(3 votes)
- You missed the 2nd minus sign.
-2/(-6) = +1/3 because a negative divided by a negative = a positive.
Hope this helps.(4 votes)
- So, the quadratic formula can be used on ANY quadratic equation, but isolating the square, the zero-product property, and completing the square cannot. So, is it efficient just to use the quadratic formula every time?(3 votes)
- You can use the quadratic formula for all quadratic equations, but sometimes it's quicker to just factor it if it's a simple equation.(4 votes)
We're asked to solve the quadratic equation, negative 3x squared plus 10x minus 3 is equal to 0. And it's already written in standard form. And there's many ways to solve this. But in particular, all solve it using the quadratic formula. So let me just rewrite it. We have negative 3x squared plus 10x minus 3 is equal to 0. And actually, I'll solve it twice using the quadratic formula to show you that as long as we manipulated this in the valid way, the quadratic formula will give us the exact same roots or the exact same solutions to this equation. So in this form right over here, what are our ABCs? Let's just remind ourselves what the quadratic formula even is actually. That's a good place to start. The quadratic formula tells us that if we have a quadratic equation in the form ax squared plus bx plus c is equal to 0, so in standard form, then the roots of this are x are equal to negative b plus or minus the square root of b squared minus 4ac, all of that over 2a. And this is derived from completing the square in a general way. So it's no magic here, and I've derived it in other videos. But this is the quadratic formula. This is actually giving you two solutions, because you have the positive square root here and the negative square root. So let's apply it here in the case where-- in this case, a is equal to negative 3, b is equal to 10, and c is equal to negative 3. So applying the quadratic formula right here, we get our solutions to be x is equal to negative b. b is 10. So negative b is negative 10 plus or minus the square root of b squared. b is 10. So b squared is 100 minus 4 times a times c. So minus 4 times negative 3 times negative 3. Let me just write it down. Minus 4 times negative 3 times negative 3. All of that's under the radical sign. And then all of that is over 2a. So 2 times a is negative 6. So this is going to be equal to negative 10 plus or minus the square root of 100 minus-- negative 3 times negative 3 is positive 9. Positive 9 times 4 is positive 36. We have a minus sign out here. So minus 36. All of that over negative 6. This is equal to 100 minus 36 is 64. So negative 10 plus or minus the square root of 64. All of that over negative 6. The principal square root of 64 is 8. But we're taking the positive and negative square root. So this is negative 10 plus or minus 8 over negative 6. So if we take the positive version, we say x could be equal to-- negative 10 plus 8 is negative 2 over negative 6. So that was taking the plus version. That's this right over here. And negative 2 over negative 6 is equal to 1/3. If we take the negative square root, negative 10 minus 8-- So let's take negative 10 minus 8. That would be x is equal to-- negative 10 minus 8 is negative 18. And that's going to be over negative 6. Negative 18 divided by negative 6 is positive 3. So the two roots for this quadratic equation are positive 1/3 and positive 3. And I want to show you the we'll get the same answer, even if we manipulate this. Some people might not like the fact that our first coefficient here is a negative 3. Maybe they want a positive 3. So to get rid of that negative 3, they can multiply both sides of this equation times negative 1. And then if you did that, you would get 3x squared minus 10x plus 3 is equal to 0 times negative 1, which is still equal to 0. So in this case, a is equal to 3, b is equal to negative 10, and c is equal to 3 again. And we could apply the quadratic formula. We get x is equal to negative b. b is negative 10. So negative negative 10 is positive 10, plus or minus the square root of b squared, which is negative 10 squared, which is 100, minus 4 times a times c. a times c is 9 times 4 is 36. So minus 36. All of that over 2 times a. All of that over 6. So this is equal to 10 plus or minus the square root of 64, or really that's just going to be 8. All of that over 6. If we add 8 here, we get 10 plus 8 is 18 over 6. We get x could be equal to 3. Or if we take the negative square root or the negative 8 here, 10 minus 8 is 2. 2 over 6 is 1/3. So once again, you get the exact same solutions.