Question

What is the smallest positive value of the expression 2z^2 + 40 if z = 3x + 42?

a) 122
b) 142
c) 162
d) 202

Answer 1

Therefore, the smallest positive value of the expression 2z^2 + 40 if z = 3x + 42 is 3392 + 40 = 3432.

Step-by-step explanation:

Let's start by substituting the given expression for z into the expression 2z^2 + 40:

2z^2 + 40 = 2(3x + 42)^2 + 40

Expanding the square:

2z^2 + 40 = 2(9x^2 + 24x + 1764) + 40

Distributing the 2:

2z^2 + 40 = 18x^2 + 48x + 3528 + 40

Combining like terms:

2z^2 + 40 = 18x^2 + 48x + 3568

To find the smallest positive value of this expression, we need to minimize the value of 18x^2 + 48x + 3568.

Completing the square for the quadratic term 18x^2 + 48x, we get:

18x^2 + 48x + 3568 = 18(x^2 + 8/3x) + 3568

To complete the square, we add and subtract (8/3)^2/4 = 16/9 inside the parentheses:

18(x^2 + 8/3x + 16/9) + 3568 - 18(16/9)

18(x + 4/3)^2 + 3392

Since the square of a real number is always non-negative, the smallest value of 18(x + 4/3)^2 is 0. Therefore, the smallest possible value of 18x^2 + 48x + 3568 is 3392.

Answer 2

Substitute the given value of z into the expression, find the vertex of the parabolic expression to obtain the smallest positive value.

Step-by-step explanation:

The smallest positive value of the expression 2z^2 + 40 can be found by substituting the given value of z = 3x + 42 into the expression. So, we have:

2(3x + 42)^2 + 40

Simplifying further:

2(9x^2 + 252x + 1764) + 40

18x^2 + 504x + 3528 + 40

18x^2 + 504x + 3568

The smallest positive value occurs when the x-value yields the vertex of the parabolic expression. The x-coordinate of the vertex can be found using the formula -b/2a, where a = 18 and b = 504. Substituting these values gives:

x = -504/(2*18) = -504/36 = -14

Therefore, the smallest positive value of the expression 2z^2 + 40 is obtained when x = -14. Substituting this value into the expression gives:

2(-14)^2 + 40 = 2(196) + 40 = 392 + 40 = 432

So, the correct answer is d) 202.