1) without solving the equation show that the solution to the above equation lies in the interval - 1(< and eqaul to) x (

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Sorry it must have not related the rest. 1)without solving the equation show that the solution to the above equation lies in the interval - 1(< and eqaul to) x (< and equal to) 5. 2)Solve the equation. 3) without any further calculations solve the equation √ 5-x = x+1

Thanks for that. I have edited my answer.

√(5-x)=x+1.

(1) When x=-1, we get √6, and x+1=0, and √6>0.

When x=5, we get 0 (LHS) and x+1=6, and 0<6.

Note that we have a change of sign implying that 0 comes somewhere in between. That means there must be a value of x (=c) between -1 and 5 such that √(5-c)=c+1.

(2) Square both sides:

5-x=x²+2x+1, x²+3x-4=0=(x+4)(x-1).

There appear to be two roots, x=1 and x=-4.

(3) We know from (1) that the solution is between -1 and 5, so x=1 must be the solution, because it lies between these two values.

(Why isn’t x=-4 a solution? Plug x=-4 into the original equation and we get:

√9=-3. Certainly, (-3)²=9, but the question specifies an implied positive square root and -3 is negative, so we throw out x=-4 as a solution.)

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