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For which values of
b will
bx+x^(2)+35 have a minimum value of 19 ? If there is more than one answer, separate them using a comma.

For which values of $b$ will $b x+x^{2}+35$ have a minimum value of $19$ ? If there is more than one answer, separate them using a comma.

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Complex conjugate theorem

Full solution

Q. For which values of

b

will

b x+x^{2}+35

have a minimum value of

19

? If there is more than one answer, separate them using a comma.

Quadratic Function Form: The given function is a quadratic function of the form $f(x) = ax^2 + bx + c$ . To find the minimum value of a quadratic function, we can use the vertex form of a parabola, which is given by $f(x) = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the parabola. The minimum value of the function is $k$ when $a > 0$ .
Comparison with Standard Form: The given function is $f(x) = x^2 + bx + 35$ . We can compare this with the standard form of a quadratic function, which is $f(x) = ax^2 + bx + c$ . Here, $a = 1$ , $b$ is the coefficient we need to find, and $c = 35$ .
Vertex Calculation: The vertex of the parabola represented by the quadratic function is given by the formula $h = -\frac{b}{2a}$ . Since $a = 1$ , we have $h = -\frac{b}{2}$ .
Minimum Value Determination: The minimum value of the function occurs at the vertex. The y-coordinate of the vertex, $k$ , is the minimum value of the function. We are given that the minimum value is $19$ , so $k = 19$ .
Substitution and Equation Setup: To find the minimum value $k$ , we substitute $x = h$ into the function $f(x) = x^2 + bx + 35$ and set it equal to $19$ . This gives us the equation $(h)^2 + b(h) + 35 = 19$ .
Solving for b: Substituting $h = -\frac{b}{2}$ into the equation, we get $\left(-\frac{b}{2}\right)^2 + b\left(-\frac{b}{2}\right) + 35 = 19$ . Simplifying this, we get $\frac{b^2}{4} - \frac{b^2}{2} + 35 = 19$ .
Final Steps: Simplifying further, we combine like terms to get $(b^2/4) - (2b^2/4) + 35 = 19$ , which simplifies to $(-b^2/4) + 35 = 19$ .
Conclusion: Subtracting $35$ from both sides of the equation, we get $(-b^2/4) = 19 - 35$ , which simplifies to $(-b^2/4) = -16$ .
Final Answer: Multiplying both sides of the equation by $-4$ to solve for $b^2$ , we get $b^2 = 64$ .
Final Answer: Multiplying both sides of the equation by $-4$ to solve for $b^2$ , we get $b^2 = 64$ . Taking the square root of both sides of the equation, we find that $b$ can be either $8$ or $-8$ , since both numbers squared give $64$ .

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