Question

X^2+10x+19=0 complete the square PLEASE EXPLAIN

Answer 1

Solve the quadratic equation by completing the square: x²+10x+19=0

Form:
`(x+5)^2=6`

Solutions:
`x=-5+√(6) , -5-√(6)`

Step-by-step explanation below with work↓

First, you are going to have to bring the 19 to the other side. When you bring a number to another side their sign changes. In this case, 19 becomes -19. Leave a space though after the 10x as you are going to be inserting something later

.
`x^2+10x+19=0\\\\x^2+10x=-19`

Second, you are going to find what half of 10 in 10x is. In this case, it is 5, after you are going to square the 5 to get 25. Keep in mind the 5 as it is going to help when we factor later.

`(1)/(2) (10)=5\\\\5^2=25`

Third, we are going to add 25 to both sides.

`x^2+10x+25=-19(+25)\\\\x^2+10x+25=6`

Fourth, we need to factor out the equation. Remember the 5? This is where it comes into place, 5 squared is 25. Then we need to get rid of the power of 2, by that we square root but whatever we do to the left we do to the right so we end up square rooting both sides. A square root cancels out the power of two making it disappear. To the 6 you have to add the plus or minus sign also. This is the plus or minus symbol ±.

`x^2+10x+25=6\\\\(x+5)^2=6\\\\√((x+5)^2) =√(6\\)\\\\ x+5=+-√(6)`

Fifth, you cannot break down the 6 further so you are going to leave it alone. If we were able to break down the 6 further they would have to be divided perfectly by a perfect square. Then we have to isolate the x, to do that we bring over the 5, and its sign changes making it -5.

`x+5=+-√(6) \\x=-5+-√(6)`

Sixth, since the answer is x = -5±√6 you are going to have two solutions.

`x= -5+√(6) , -5-√(6)`

Answer 2

Explanation

In the form ax^2+bx+c

a x c should equal two factors of b

However, in this equation

1 x 19 has no factors that = 10

so you must use the Quadratic Formula which is,

x =
`(-b+-√(b^2-4ac) )/(2a)`

so the answer should be

x =
`(-10+-√(10^2-4(1)(19)) )/(2(1))`

x =
`(-10+-√(24) )/(2)`

x =
`(-10+2√(6) )/(2)` and x =
`(-10-2√(6) )/(2)`