Find possible values of the constant p if the expression px²+12x+4p+px is a perfect square

Answer 1

To determine the possible values of the constant p for which the expression px² + 12x + 4p + px is a perfect square, we can rewrite the expression as:

px² + px + 12x + 4p

Grouping the terms:

p(x² + x) + 12(x + p)

Factoring out x² + x:

p(x(x + 1)) + 12(x + p)

To make this expression a perfect square, we need to complete the square for both terms.

For p(x(x + 1)), we need to add the square of half the coefficient of the x term, which is (1/2)^2 = 1/4. However, we also need to consider the factor of p in front, so we actually need to add p(1/4):

p(x(x + 1) + 1/4).

For 12(x + p), we need to add the square of half the coefficient of the x term, which is (12/2)^2 = 36. However, we also need to consider the factor of p in front, so we actually need to add p(36):

12(x + p) + p(36).

Combining these terms:

p(x(x + 1) + 1/4) + 12(x + p) + p(36).

To make this expression a perfect square, we need to set these two terms equal to each other:

x(x + 1) + 1/4 = 12.

Expanding and rearranging:

x² + x + 1/4 - 12 = 0.

x² + x - 47/4 = 0.

To solve this quadratic equation, we can use the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a).

In this case, a = 1, b = 1, and c = -47/4. Plugging in these values:

x = (-(1) ± √((1)² - 4(1)(-47/4))) / (2(1)).

Simplifying:

x = (-1 ± √(1 + 47/4)) / 2.

x = (-1 ± √(51/4)) / 2.

x = (-1 ± √51/2) / 2.

Since p is a constant, it can be any real number. Therefore, there are no specific values for p that would make the expression a perfect square.

Answer 2

px^2+12x+4p+px = px^2 + (p+12)x + 4p = p(x^2 + (1 + 12/p)x + 4)
We need p to be a perfect square, and it is clear by inspection that if p=4, we have
4(x^2 + 4x + 4) = (2(x+2))^2

Answer 3

Apologies for the previous incorrect response. You are correct that we can rewrite the expression as px² + (p+12)x + 4p.

To make this expression a perfect square, we need the coefficient of x to be twice the square root of the constant term. In this case, the constant term is 4p.

Therefore, (p+12) = 2√(4p)

Simplifying the equation:

p + 12 = 2√(4p)

p + 12 = 4√p

(√p)^2 - 4√p + 12 = 0

Letting z = √p, we can rewrite the equation as:

z^2 - 4z + 12 = 0

Applying the quadratic formula:

z = (4 ± √(4² - 4(1)(12)))/2

z = (4 ± √(16 - 48))/2

z = (4 ± √(-32))/2

Since we can't take the square root of a negative number, there are no possible values of p that make the expression a perfect square.