Solve the given equation:
ln(x + 4) + ln(x + 2) = 2
ps - we aren't allowed calculators
My ans:
I got as far as
ln [(x+4)(x+2)] = 2
ln (x^2+6x+8) = 2
And then I am lost, whatever I do ends up with no answer.
I also rewrote the above so that it is
e^2 = x^2+6x+8
But of course that made it worse :(
I am also stuck just as you are.
i also got the same equation:
ln [(x+4)(x+2)] = 2
ln (x^2+6x+8) = 2
x^2+6x+8 = e^2
x^2+6x+(8-e^2)=0
thus, you have no choice but to use quadratic formula: (where a=1, b=6, c=8-e^2)
[-6+-sqrt(6^2-4(1)(8-e^2))]/2(1)
[-6+-sqrt(36-32+4e^2)]/2
[-6+-sqrt(4+4e^2)]/2 *factor out 4
[-6+-2(sqrt{1+e^2})]/2
therefore, x is equal to:
-3+sqrt(1+e^2) and -3-sqrt(1+e^2)
so there,, =)
To solve the equation ln(x + 4) + ln(x + 2) = 2, you're on the right track. Here's how to proceed:
1. Start with the equation ln[(x + 4)(x + 2)] = 2.
2. To simplify the equation, you can convert it from the logarithmic form to the exponential form. Recall that for any positive numbers a and b, ln(a) = b is equivalent to e^b = a, where e is the natural logarithm base (approximately 2.71828).
3. Applying this transformation to your equation, you can rewrite it as e^2 = (x + 4)(x + 2).
4. Expand the right side of the equation by multiplying (x + 4)(x + 2):
e^2 = x^2 + 2x + 4x + 8.
5. Simplify the equation further:
e^2 = x^2 + 6x + 8.
Now, to solve the equation e^2 = x^2 + 6x + 8, you can rearrange it to be a quadratic equation:
1. Move all terms to one side of the equation:
x^2 + 6x + 8 - e^2 = 0.
2. Since you are not allowed to use a calculator, you can leave it in this form and attempt to factorize the quadratic equation. You are looking for two numbers that multiply to give 8 and sum to 6. However, this equation may not factor nicely with whole numbers, so you can try the quadratic formula instead:
x = (-b ± √(b^2 - 4ac)) / (2a),
where a, b, and c are the coefficients of the quadratic equation x^2 + bx + c = 0.
In this case, the coefficients are:
a = 1, b = 6, c = 8 - e^2.
Plugging these values into the quadratic formula will give you the solutions for x.