simplify
(x+3)^2= 19
use foil
(x+3)(x+3)=19
x^2+6x+9=19
x^2+ 6x=10
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To simplify the equation (x+3)^2 = 19, we need to first expand the square on the left side of the equation.
Step 1: Expanding the square
(x+3)^2 = (x+3)(x+3) = x(x+3) + 3(x+3) = x^2 + 3x + 3x + 9 = x^2 + 6x + 9
So, the equation becomes: x^2 + 6x + 9 = 19
Step 2: Rearranging the equation
To simplify further, we need to move the constant term (9) to the right side of the equation. Subtracting 9 from both sides yields:
x^2 + 6x = 19 - 9
Simplifying this expression gives:
x^2 + 6x = 10
Step 3: Solving the equation
To solve this quadratic equation, we can either factor it, complete the square, or use the quadratic formula. Let's use the quadratic formula in this case.
The quadratic formula states that for an equation in the form ax^2 + bx + c = 0, the solution for x is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = 1, b = 6, and c = -10. Plugging these values into the quadratic formula, we get:
x = (-6 ± √(6^2 - 4(1)(-10))) / (2(1))
x = (-6 ± √(36 + 40)) / 2
x = (-6 ± √76) / 2
x = (-6 ± √(4 * 19)) / 2
x = (-6 ± 2√19) / 2
Simplifying further:
x = -3 ± √19
Therefore, the solutions to the equation (x+3)^2 = 19 are x = -3 + √19 and x = -3 - √19.