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Quadratic equations word problem: triangle dimensions

Sal solves a geometry problem using a quadratic equation. Created by Sal Khan and Monterey Institute for Technology and Education.

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  • purple pi purple style avatar for user Joaquin Lerner
    At when he says "let me make it clear... lets do it this way"
    How did he know to multiply the 1/2 with the b before distributing?
    Is that part of the order of operations? Are you supposed to do any multiplications that come before the parentheses first before distributing always?
    (25 votes)
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  • purple pi purple style avatar for user mihir.borkar
    How do you do factoring by grouping?
    (8 votes)
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    • old spice man green style avatar for user Petrie (Peter S. Asiain III)
      I'll explain how to do factoring by grouping but first be aware that.....
      The quadratic equation is in the form of...........
      Ax²+Bx+C=0
      Where:
      (note:coefficient of means the number beside)
      A=the coefficient of x²
      B=the coefficient of x
      C=the constant term(or the term that doesn't have an x beside it)
      And we want to find two coefficients of x (q and w) to replace B which when you.......
      multiply them they equal = A*C =q*w
      add them they equal = A+B =q*w

      so from..... Ax²+Bx+C=0
      you'll get.... Ax²+qx+wx+C=0
      you'll group them to.. (Ax²+qx)+(wx+C)=0
      you'll further simplify it with real numbers...

      Okay lets have an example so that we can understand it even more.
      Lets say we want to factor an equation the form of Ax²+Bx+C=0
      like 2x²+13x+20=0
      We know that our A=2, B=13 and C=20
      So we want to find two coefficients of x (q and w) to replace B which when you.......
      multiply them they equal = A*C =q*w= 2*20=40
      add them they equal = A+B =13=q*w =8*5
      Now we know that our q and w are..... 8 and 5 because the satisfy all the conditions we are looking for.
      Lets just plug in their values to..
      2x²+qx+wx+20=0
      2x²+8x+5x+20=0
      (2x²+8x)+(5x+20)=0 ( I grouped them apropriately so that I can factor out)
      2x(x+4)+5(x+4)=0 ( I factored out a 2 and 5)
      (x+4) (2x+5)=0 ( I factored out the (x+4) from 2x and 5)
      (19 votes)
  • leaf green style avatar for user Emily Witanga
    What I am still so confused on how to solve problems using quadratic equations?
    (6 votes)
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  • female robot grace style avatar for user Sonata
    So are all measurement/geometry problems using quadratics like the one in the video going to have a negative answer and a positive answer? If not, how will you know which one is the correct answer? Will one give you a different answer from the other when you put the value in the original equation?
    (3 votes)
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    • duskpin ultimate style avatar for user Patrick Stetz
      With respect to Geometry, the right answer will always be positive, since it's impossible to have negative area, length, volume, etc. So if there's one positive answer and one negative answer, the correct one will be positive.

      The triangle problem that Sal was doing in the video will always have one positive and one negative answer. This is because the area will always be positive making "c" (ax^2+bx+c) always negative. When "c" is negative there is only one positive solution and one negative solution.

      It's also important to remember that sometimes the quadratic equation will give two positive answers or two negative answers (if you plug in x^2-5x+6 to the quadratic equation, you get an example of two positive answers, while x^2+5x+6 gives you an example of two negative answers.)

      Hope that helps
      (5 votes)
  • purple pi purple style avatar for user annika.brown143
    What if you had 35n^2 + 22n + 3 = 0?
    What should I do?
    (3 votes)
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    • duskpin ultimate style avatar for user Patrick Stetz
      3 only has two factors, so it is inevitable that the solution will look something like this...
      (_n + 3)(_n + 1)
      Now, all we have to deal with are the factors of 35 and there are four possibilities (1 & 35, 35 & 1, 5 & 7, and 7 & 5), so it shouldn't be too hard to check each one.
      1.) (1n + 3)(35n + 1)
      35n^2 + n + 105n + 3
      35n^2 + 106n + 3 =/= 35n^2 + 22n + 3
      This does not work, they are not equal
      2.) (35n + 3)(n + 1)
      35n^2 + 35n + 3n + 3
      35n^2 + 38n + 3 =/= 35n^2 + 22n + 3
      This does not work
      3.) (5n + 3)(7n + 1)
      35n^2 + 5n + 21n + 3
      35n^2 + 26n + 3 =/= 35n^2 + 22n + 3
      This does not work, so it must be the last possibility
      4.) (7n + 3)(5n + 1)
      35n^2 + 7n + 15n + 3
      35n^2 +22n + 3 = 35n^2 + 22n + 3
      They match so (7n + 3)(5n + 1) is the correct factorization.

      To finish the problem...
      (7n + 3)(5n +1) = 0
      This means that either 7n + 3 = 0 or 5n + 1 = 0
      7n + 3 = 0
      7n = -3
      n = -3/7
      5n + 1 = 0
      5n = -1
      n = -1/5
      n = -3/7 or -1/5
      (3 votes)
  • spunky sam green style avatar for user John
    are there any quadratics that can't be factored?
    (2 votes)
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    • male robot hal style avatar for user danny
      Try x^2 + 2x + 2.

      Any quadratic of the form ax^2 + bx + c can be solved using the formula ( -b +/- sqrt(D) )/2a with D= b^2-4*a*c. However, if D is less than zero it cannot be solved regularly. This requires the introduction if the imaginary number i = sqrt(-1). I think this allows us to factor all quadratics.
      So short answer: yes. Long short answer: no.
      (5 votes)
  • eggleston green style avatar for user vincenzo
    At , Sal says the area is 30 inches SQUARED equal to 1/2 the base times height, yet he wrote it as:


    30 = 1/2*b(b-4)


    Why didn't he write it as 30^2 like the problem said?:


    30^2 = 1/2*b(b-4)
    (2 votes)
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  • female robot amelia style avatar for user Kiana  Reese
    At about in the video, at (b squared/2-2b) times 2, where did he get the 2 that he multiplied on both sides of the equation?
    (3 votes)
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    • leaf orange style avatar for user Mikhail Chupakhin
      On the right side of the expression, we have 1/2*b*(b-4). Thus, we will have 1/2*b^2-2 after multiplication. Since we are looking for "b" and not a fractional part of it, it is quite convenient to multiply the expression by 2 to get rid of 1/2. However, we cannot multiply only one side of the equation and we need to multiply 30 by 2 as well. 2*30 = 2(b^2/2 - 2b)
      (2 votes)
  • hopper jumping style avatar for user Ty
    Why can't a trinagle's base be negative?
    (2 votes)
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  • mr pants pink style avatar for user sunny sun
    at , he said let's multiply both sides by two to get rid of the 2 in the fraction, and proceeded to multiply the whole equation by two. But why couldn't you just simply multiplied the fraction by two, it would've gotten rid of the fraction anyways.
    (2 votes)
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    • stelly blue style avatar for user Kim Seidel
      You can't just multiply one term in an equation by 2. To create an equivalent equation, you must multiply the entire equation by 2. Do a simple example that contains just numbers.
      Consider: 1 + 5 = 6. The 2 sides are currently equal.
      -- Multiply just 1 by 2. You get: 2 + 5 = 6. The 2 sides are no longer equal.
      -- Multiply the entire equation by 2. You get: 2 + 10 = 12. The 2 sides are still equal.

      Hope this helps.
      (3 votes)

Video transcript

The height of a triangle is four inches less than the length of the base. The area of the triangle is 30 inches squared. Find the height and base. Use the formula area equals one half base times height for the area of a triangle. OK. So let's think about it a little bit. We have the-- let me draw a triangle here. So this is our triangle. And let's say that the length of this bottom side, that's the base, let's call that b. And then this is the height. This is the height right over here. And then the area is equal to one half base times height. Now in this first sentence they tell us at the height of a triangle is four inch is less than the length of the base. So the height is equal to the base minus 4. That's what that first sentence tells us. The area of the triangle is 30 inches squared. So if we take one half the base times the height we'll get 30 inches squared. Or we could say that 30 inches squared is equal to one half times the base, times the height. Now instead of putting an h in for height, we know that the height is the same thing as 4 less than the base. So let's put that in there. 4 less than the base. And then let's see what we get here. We get-- let me do this in yellow. We get 30 is equal to one half times-- let's distribute the b-- times b, let me make it clear. So let's do it this way. Times b over 2, times b, minus 4. I just multiplied the one half times the b. Now let's distribute the b over 2. So 30 is equal to b squared over 2, be careful. b over 2 times b is just b squared over 2. And then b over 2, times negative 4 is negative 2b. Now just to get rid of this fraction here let's multiply both sides of this equation by 2. So let's multiply that side by 2. And let's multiply that side by 2. On the left hand side you get 60. On the right hand side 2 times b squared over 2 is just b squared. Negative 2b times 2 is negative 4b. And now we have a quadratic here. And the best way to solve a quadratic-- we have a second degree term right here-- is to get all of the terms on one side of the equation, having them equal 0. So let's subtract 60 from both sides of this equation. And we get 0 equal to b squared, minus 4b, minus 60. And so what we need to do here is just factor this thing right now, or factor it. And then, no-- if I have the product of some things, and that equals 0, that means that either one or both of those things need to be equal to 0. So we need to factor b squared, minus 4b, minus 60. So what we want to do, we want to find two numbers whose sum is negative 4 and whose product is negative 60. Now, given that the product is negative, we know there are different signs. And this tells us that their absolute values are going to be four apart. That one is going to be four less than the others. So you could look at the products of the factors of 60. 1 and 60 are too far apart. Even if you made one of the negative, you would either get positive 59 as the sum or negative 59 as the sum. 2 and 30, still too far apart. 3 and 20, still too far apart. If you had made one negative you'd either get negative 17 or positive 17. Then you could have 4 and 15, still too far apart. If you made one of them negative, their sum would be either negative 11 or positive 11. Then you have 5 and a 12, still seems too far apart from each other. One of them is negative, then you either have their sum being positive 7 or negative 7. Then you have 6 and 10. Now this looks interesting. They are four apart. So if we make-- and we want the larger absolute magnitude number to be negative so that their sum is negative. So if we make it 6 and negative 10 their sum will be negative 4, and their product is negative 60. So that works. So you could literally say that this is equal to b plus 6, times b, minus 10. b plus the a, plus b minus the b. And let me be very careful here. This b over here, I want to make it very clear, is different than the b that we're using in the equation. I just used this b here to say, look, we're looking for two numbers that add up to this second term right over here. It's a different b. I could have said x plus y is equal to negative 4, and x times y is equal to negative 60. In fact, let me do it that way just so we don't get confused. So we could write x plus y is equal to negative 4. And then we have x times y is equal to negative 60. So we have b plus 6, times b plus y. x is 6, y is negative 10. And that is equal to 0. Let's just solve this right here. And then we'll go back and show you. You could also factor this by grouping. But just from this, we know that either one of these is equal to zero. Either b plus 6 is equal to 0, or b minus 10 is equal to 0. If we subtract 6 from both sides of this equation, we get b is equal to negative 6. Or if you add 10 to both sides of this equation, you get b is equal to 10. And those are our two solutions. You could put them back in and verify that they satisfy our constraints. Now the other way that you could solve this, and we're going to get exact same answer. Is you could just break up this negative 4b into its constituents. So you could have broken this up into 0 is equal to b squared. And then you could have broken it up into plus 6b, minus 10b, minus 60. And then factor it by grouping. Group these first two terms. Group these second two terms. Just going to add them together. The first one you could factor out a b. So you have b times b, plus 6. The second one you can factor out a negative 10. So minus 10 times b, plus 6. All that's equal to 0. And now you can factor out a b plus 6. So if you factor out a b plus 6 here, you get 0 is equal to b minus 10, times b, plus 6. We're literally just factoring out this out of the expression. You're just left with a b minus 10. You get the same thing that we did in one step over here. Whatever works for you. But either way, the solutions are either b is equal to negative 6, or b is equal to 10. And we have to be careful here. Remember, this is a word problem. We can't just state, oh b could be negative 6 or b could be 10. We have to think about whether this makes sense in the context of the actual problem. We're talking about lengths of triangles, or lengths of the sides of triangles. We can't have a negative length. So because of that, the base of a triangle can't have length of negative 6. So we can cross that out. So we actually only have one solution here. Almost made a careless mistake. Forgot that we were dealing with the word problem. The only possible base is 10. And let's see, they say find the height and the base. Once again, done. So the base we're saying is 10. The height is four inches less. It's b minus 4. So the height is 6. And then you can verify. The area is 6 times 10 times one half, which is 30.