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Solve by completing the square: Integer solutions

We can use the strategy of completing the square to solve quadratic equations. If the solutions are integers, we can solve by factoring as well. Created by Sal Khan.

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• why would we divide the 8 and then do 4² i do not get this at all.
• Square of binomials (x+a)^2, when you expand them, always have the form x^2+2ax+a^2. What Sal did was divide the coefficient of the second term (equivalent to 2a of the 2ax), 8 in this example, by 2 to get a, which is 4. He then squared it (4^2) to get the last term, a^2.
• Are there more simple ways to do this?
(I mean, this is pretty simple, and I understand it, but I'm just am curious if there are easier ways)
• By subtracting 85, you get x^2-8x-84=0, so factor by finding two numbers that multiply to be 84 and subtract to get 8, so try a few until you find -14+6 so factors are (x-14)(x+6)=0. By zero product rule, x=14 or x=-6. Or you could put into a graphing calculator and find x intercepts. Simple may be a matter of opinion. Completing the square and quadratic formula work well with solutions that include a radical.
• Still wobbly on the fact that the sqrt(100) is +/- 10 instead of "just" 10. Help :).
• To find the square root of 100, we need to find a number that, when multiplied by itself, gives us 100. The reason why it is +/- 10 is that both positive 10 and negative 10 give us 100.

-10 * -10 = 100
10 * 10 = 100

This works with all square roots (I think).
• At (x-4)^2 - 100.
can be rewritten as (x-4)^2 -(10)^2
here, we can use the difference of squares as well
(x-4+10) (x-4-10) = 0
(x+6) (x-14) =0
So, x roots are -6 and 14.
I apologize if it seems misleading :)
• Why does x^2-8x+16 equal to (x-4)^2 at
• Going backwards, (x-4)(x-4)=x(x-4)-4(x-4)=x^2-4x-4x-+16=x^2-8x+16. This is an example of a perfect square polynomial, if you have -8/2=-4 and (-4)^2=16, then it fits the perfect square.
• I'm not getting how he got (x-4)^2
(1 vote)
• He got (x-4)^2 because (x-4)^2 is technically (x-4)(x-4). (x-4) times (x-4) is x^2 - 4x - 4x + 16 using FOIL, and x^2 - 4x - 4x + 16 can be simplified to x^2 - 8x + 16, which is what Sal got to make the equation a complete square.

Hope this helps!
• Why don't we take the plus or minus square root of the x-4?
• Lets suppose you could add the ± on both sides of the equation. This would create 4 possibilities:
(x-4) = 10, (x-4)=-10, -(x-4)=10 and -(x-4)=-10. Looking at the second 1, divide by negative 1 to get (x-4)=-10 and you are back at the second one. Doing the same thing on the 4th, you get (x-4)=10 which is the same as the first. So you really only need to do it on one side, the other side will just repeat what you already have.
• always she if the quadratic equations is on standar form "ax2+bx+c"
• Sort of, the standard form is y=ax^2+bx+c or f(x)=ax^2+bx+c. To find x intercepts (roots, zeroes, or solutions), you set y=0 to get ax^2+bx+c=0. Solving this by completing the square gives the quadratic formula.
• What if the other side of the equation has an X? My question asks, X^2 - 4x + 4 = 2x