Someone help me!!
square root = sqrt
(5x^2-5x-10)sqrt(x+7) + (2x+6)sqrt(x+2) >= x^3+13x^2-6x-28
"12th grade" tells us nothing about what you need help with. Is this trig, calc, algebra, or what SUBJECT? I tutor in English and social studies and communication and journalism. I can't help you. The proper tutor may never respond to "12th grade'.
(5x^2-5x-10)√(x+7) + (2x+6)√(x+2) ≥ x^3+13x^2-6x-28 , where x ≥ -2
A quick mental inspection shows that when x = 2, the roots become exact.
Could it be ? could it be?
testing for x = 2
LS = (20-10-10)√9 + (4+6)√4
= 0 + 20
= 20
RS = 8 + 4(13) - 12 - 28
= 20
so x = 2 , at the equality
Well, that was a "lucky" but logical guess .
I graphed the LS and the RS as separated functions to see where they are relative to teach other:
http://www.wolframalpha.com/input/?i=plot+y+%3D+(5x%5E2-5x-10)%E2%88%9A(x%2B7)+%2B+(2x%2B6)%E2%88%9A(x%2B2),+y++%3D+x%5E3%2B13x%5E2-6x-28+for+x+%3D+-2+to+5
testing at x = 0
LS = -10√7 + 6√2 = appr -17.98
RS = -28
So LS ≥RS at x = 0
It is highly unlikely, looking at my graphs, that there are any other solutions, so
-2 ≤ x ≤ 2
To solve this inequality, we'll need to follow a few steps:
Step 1: Simplify the expression on both sides of the inequality.
Start by expanding the square roots using the distributive property:
sqrt(5x^2 - 5x - 10)sqrt(x + 7) + sqrt(2x + 6)sqrt(x + 2) >= x^3 + 13x^2 - 6x - 28
sqrt((5x^2 - 5x - 10)(x + 7)) + sqrt((2x + 6)(x + 2)) >= x^3 + 13x^2 - 6x - 28
Step 2: Combine like terms.
There are no like terms on either side of the inequality, so we can move on to the next step.
Step 3: Isolate the square roots.
To get rid of the square roots, we need to square both sides of the inequality. However, when squaring both sides, we need to be careful because this can introduce extraneous solutions. Therefore, after squaring, we must check for any potential extraneous solutions.
(Squaring both sides)
(5x^2 - 5x - 10)(x + 7) + 2sqrt((5x^2 - 5x - 10)(x + 7))(2x + 6) + (2x + 6)^2(x + 2) >= (x^3 + 13x^2 - 6x - 28)^2
Simplify the equation.
(5x^2 - 5x - 10)(x + 7) + 4sqrt((5x^2 - 5x - 10)(x + 7))(x + 3) + (2x + 6)^2(x + 2) - (x^3 + 13x^2 - 6x - 28)^2 >= 0
Step 4: Solve the resulting equation.
At this point, we have a polynomial equation. We can simplify it by expanding and collecting like terms to obtain a quadratic equation. Then, we can solve the quadratic equation for x.
Step 5: Check for the extraneous solutions.
After solving the equation, substitute each solution back into the original inequality and check if it holds true. Any solution that makes the original inequality false should be discarded as an extraneous solution.
This process will help you solve the given inequality equation.