What is the solution of the equation? (step by step)
sqrt(2x+13) -5 =x

Step 1: Isolate the radical expression

+4 is not in the radical but is on the same side. I need it to not be there. So I will add 4 to both sides to get..

sqrt(2x+3) = x - 4

Step 2: Square both sides!

Now I need to get rid of the square root sign. What is the opposite of square rooting something? Squaring it! So square both sides to get...

2x+3 = (x - 4)^2

Don't forget to square the ENTIRETY of both sides.. USE PARENTHESES! Step 3: Combine like terms and solve for x.

2x + 3 = (x - 4)^2

So I first need to distribute out the (x - 4)^2 (x-4)(x-4) to get x^2 - 8x + 16.

So...

2x + 3 = x^2 - 8x + 16.

This is a quadratic equation so I will need to set it equal to zero and solve by factoring, graphing, or quadratic formula. I am going to subtract a 2x from both sides and a 3 from both sides to get...

0 = x^2 - 10x +13

I solved by graphing the quadratic equation and I got x={1.536, 8.464}

Step 4: Check for extraneous solutions.

Sometimes by doing all the tricky algebra you get extra answers that don't actually work. So it's important to check your solutions to see which ones do and do not work. Check by plugging them in to the ORIGINAL equation...

sqrt(2*1.535 + 3) + 4 ?=? 1.535

Which gives you...

6.464 = 1.535 FALSE FALSE FALSE.. This must be an extra answer... or extraneous

sqrt(2*8.464 + 3) + 4 ?=? 8.464

Which gives you...

8.464 = 8.464 TRUE!!!

So x=8.464 is the actual solution!

d.) yes, because the volume of the cone is greater than the volume of the sphere

step-by-step explanation:

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