Answer:
x = (-10 + sqrt(180)) / 8
= -1.25 + 0.75 * sqrt(5)
x = (-10 - sqrt(180)) / 8
= -1.25 - 0.75 * sqrt(5)
Explanation:
To solve the given equation, we need to combine like terms and isolate the variable on one side of the equation. The given equation is:
2x(x+4)+7=(x+8)+2x(x+1)+12
We can start by expanding the parentheses on the left-hand side to obtain:
2x^2 + 8x + 7 = x + 8 + 2x^2 + 2x + 12
We can then combine like terms on both sides of the equation to get:
4x^2 + 10x - 5 = 0
To solve for x, we can use the quadratic formula, which states that the solutions of the equation ax^2 + bx + c = 0 are given by:
x = (-b +/- sqrt(b^2 - 4ac)) / 2a
In the given equation, we have a = 4, b = 10, and c = -5, so the solutions are:
x = (-10 +/- sqrt(10^2 - 4 * 4 * (-5))) / 2 * 4
= (-10 +/- sqrt(100 + 80)) / 8
= (-10 +/- sqrt(180)) / 8
This gives us two solutions:
x = (-10 + sqrt(180)) / 8
= -1.25 + 0.75 * sqrt(5)
x = (-10 - sqrt(180)) / 8
= -1.25 - 0.75 * sqrt(5)
These are the two solutions of the given equation. Note that we could also write the solutions in a simplified form by factoring the quadratic and using the zero product property, but the quadratic formula is generally more reliable and easier to use.