What-is-the-sum-of-all-the-solutions-of-the-equation-2x-8-2-9x-36-9-0-

Question Number 112838 by bobhans last updated on 10/Sep/20

What is the sum of all the solutions

of the equation ∣ 2 x + 8 ∣^{2} - ∣ 9 x + 36 ∣ - 9 = 0

Answered by bemath last updated on 10/Sep/20

the equation equivalent to

4 ∣ x + 4 ∣^{2} - 9 ∣ x + 4 ∣ - 9 = 0

let ∣ x + 4 ∣ = λ

\to 4 λ^{2} - 9 λ - 9 = 0

\to (4 λ + 3) (λ - 3) = 0

λ = - \frac{3}{4} (rejected), for λ = 3 (acceptable)

\to∣ x + 4 ∣= 3 \to {\begin{cases} x_{1} + 4 = - 3 \\ x_{2} + 4 = 3 \end{cases}

\Leftrightarrow x_{1} + x_{2} = - 8

Commented by MJS_new last updated on 10/Sep/20

funny : you get the right sum out of the

wrong solution

putting λ = x + 4 \Leftrightarrow x = λ - 4 we get

4 λ^{2} - 9 ∣ λ ∣ - 9 = 0

{\begin{cases} λ < 0 \land 4 λ^{2} + 9 λ - 9 = 0 \Rightarrow λ = - 3 \\ λ ⩾ 0 \land 4 λ^{2} - 9 λ - 9 = 0 \Rightarrow λ = 3 \end{cases}

\Rightarrow x_{1} = - 7 \land x_{2} = - 1

Commented by bemath last updated on 10/Sep/20

wkwkskwk \dots typo sir . thank you

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