Solve for X by completing the square

Question

x^2-8x-20=0                       and                    5x^2+24x=5

Answer 1

Completing the Square: x²+bx+(b/2)²

 

x²-8x-20=0

 

To complete the square, you first need to isolate your variables.

 

x²-8x=20

 

Next, we add half of "b" squared to both sides. In this case, half of -8 is -4.

 

x²-8x+(-4)²=20+(-4)²

 

Now, because x2-8x+16 is a perfect square trinomial, we can simplify to (x-4)²

 

(x-4)²=36

 

Next take the square root of both sides. When this happens, we get two separate answers for √36 = 6 or -6

 

x-4 = ±6

 

Finally, add 4 to both sides. 4+6 = 10 and 4-6= -2

 

x = 4 or x = -2

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For 5x²+24x = 5, the first thing we must do is divide everything by 5. This is because with completing the square your coefficient for x² must be 1.

 

5x²+24x=5

5      5    5

 

x²+ 24/5x = 1

 

Now we do the same thing as the previous problem by adding half of 24/5 squared, which would be (12/5)².

 

x²+24/5+(12/5)² = 1 + (12/5)²

 

(x+12/5)² = 1 + 144/25

(x+12/5)² = 169/25

 

When you take the square root of 169/25, you square root the numerator and denominator individually. The square root of 169 is 13 and the square root of 25 is 5, so...

 

x+12/5 = ±13/5

 

x = -12/5 ± 13/5

 

x = 1/5 or -5