My Math Forum Completing the square

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August 23rd, 2018, 12:19 AM   #1
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Completing the square

I need help fixing this equation. The problem is x^2-8x+4=0 and the answer is 4-2\|3, 4+2\|3 in words is 4 minus two square root three, 4 plus two square root three, but my answer is 4-2\|5,4+2\|5 ..why do I have a square root of five but not square root of three? Below is my work.
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Last edited by skipjack; August 23rd, 2018 at 09:26 PM.

 August 23rd, 2018, 02:06 AM #2 Member   Joined: Oct 2017 From: Japan Posts: 62 Thanks: 3 I would use an easier method to solve it.
August 23rd, 2018, 06:17 AM   #3
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Quote:
 Originally Posted by Gold Boy I need help fixing this equation. The problem is x^2-8x+4=0 and the answer is 4-2\|3, 4+2\|3 in words is 4 minus two square root three, 4 plus two square root three, but my answer is 4-2\|5,4+2\|5 ..why do I have a square root of five but not square root of three? Below is my work.
I am not sure why you are completing the square, but here goes

$x^2 - 8x + 4 = 0 \implies x^2 - 8x = -\ 4 \implies x^2 + 2 \left ( \dfrac{-\ 8}{2} \right ) * x = -\ 4 \implies$

$x^2 + 2(-\ 4)x + (-\ 4)^2 = -\ 4 + (-\ 4)^2 = 16 - 4 = 12 \implies$

$(x - 4)^2 = 12 \implies x - 4 = \pm \sqrt{12} = \pm \sqrt{4 * 3} = \pm 2 \sqrt{3} \implies$

$x = 4 \pm 2\sqrt{3}.$

Let's check.

$x = 4 \pm 2\sqrt{3} \implies x^2 - 8x + 4 = (16 \pm 2 * 4 * 2\sqrt{3}+ 4 * 3) - 8(4 \pm 2 \sqrt{3}) + 4 =$

$x = 16 + 12 + 4 - 32 \pm 16\sqrt{3} \mp 16\sqrt{3} = 32 - 32 = 0.$

EDIT: You must be very careful about signs when completing the square and when using the quadratic formula.

EDIT 2: You neglected to add 16 to BOTH sides of the equation, and then equated

$\sqrt{-\ 20}$ to $\sqrt{20}.$

So you made two basic errors.

Last edited by skipjack; August 23rd, 2018 at 09:25 PM.

 August 23rd, 2018, 02:38 PM #4 Global Moderator   Joined: May 2007 Posts: 6,684 Thanks: 658 $x^2-8x+4=(x-4)^2+k=x^2-8x+16+k$, so $k=-12$. Solution is $x=4\pm \sqrt{12}$. Thanks from topsquark and v8archie
 August 23rd, 2018, 10:08 PM #5 Global Moderator   Joined: Dec 2006 Posts: 20,285 Thanks: 1968 If possible, verify each step as you do it, so that any mistake is found immediately. Verifying a step isn't the same as checking that you did what you intended to, as your intention might have been incorrect. Thanks from topsquark and JeffM1

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