The largest y-value from the solution set of the nonlinear system is 3.78. (Correct choice: C)
What is the solution of a non-linear system
In this question we have a system of non-linear equation formed by a radical equation and a quadratic equation, which can be solved both analytically and graphically. By equalizing the two formulas we have the following equation:
√(2 · x + 4) = x² - 5 · x + 3
√(2 · x + 4) + 13 / 4 = x² - 2 · (5 / 2) · x + 25 / 4
√(2 · x + 4) + 13 / 4 = (x - 5 / 2) ²
√(2 · x + 4) = (x - 5 / 2) ² - 13 / 4
Then, we square both sides to eliminate the radical sign:
2 · x + 4 = (x - 5 / 2)⁴ - (13 / 2) · (x - 5 / 2)² + 169 / 16
2 · x + 4 = x⁴ + 4 · x³ · (- 5 / 2) + 6 · x² · (- 5 / 2)² + 4 · x · (- 5 / 2)³ + (- 5 / 2)⁴ - (13 / 2) · [x² + 2 · (- 5 / 2) · x + ( - 5 / 2)²] + 169 / 16
2 · x + 4 = x⁴ - 10 · x³ + (75 / 2) · x² - (125 / 2) · x + 625 /16 - (13 / 2) · x² + (65 / 2) · x - 325 / 8 + 169 / 16
2 · x + 4 = x⁴ - 10 · x³ + 31 · x² - 30 · x + 9
x⁴ - 10 · x³ + 31 · x² - 32 · x + 5 = 0
The real roots of this polynomial are:
x₁ ≈ 5.152, x₂ ≈ 2.862, x₃ ≈ 1.797, x₄ ≈ 0.189
Now we evaluate f(x) at each root:
y₁ ≈ 3.782, y₂ ≈ 3.118, y₃ ≈ 2.756, y₄ ≈ 2.092
The largest y-value from the solution set of the nonlinear system is 3.78. (Correct choice: C)
To learn more on nonlinear systems: https://brainly.com/question/10734771
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