What is the sum of the solutions to the equation 3x^2+7x-2 = (x+1)^2?

Question

I know how to figure it out, I don't know what to write for the sum of the solutions?  Do I see if any of the two solutions make the problem true?

Answer 1

Step 1. Use Mark's response to get 2x² + 5x - 3 = 0

 

Then use Viete's formulas to get that the sum of the solutions is -5/2.

 

Viete's formulas state that the set R={r₁, … ,r_n} of roots (real and complex) of a_nxⁿ + a_n-1x^n-1 + … + a₁x + a₀ = 0 satisfy the following:

 

There are C(n,k) subsets of R with k roots. Take the product of the roots in each subsets and add all products together to get a number S_k. Then S_k = (-1)^k a_n-k/a_n.

 

For example:

 

If ax² + bx + c = 0 has roots r₁ and r₂ then:

 

r₁ + r₂ = -b/a.

 

r₁r₂ = c/a

 

If ax³ + bx² + cx + d = 0 has roots r₁, r₂, and r₃ then:

 

r₁ + r₂ + r₃ = -b/a

 

r₁r₂ + r₁r₃ + r₂r₃ = c/a

 

r₁r₂r₃ = -d/a

 

Similarly for any degree polynomial.

Answer 2

3x² + 7x - 2 = (x + 1)²

3x² + 7x - 2 = x² + 2x + 1

2x² + 5x - 3 = 0

 

Can you solve and answer?