Find (3x^2 + 5)^2 minus what you are given.
9x^4 = (3x^2)^2
25 = 5^2
(3x^2 + 5)^2 = 9x^4 + 30x^2 + 25
To make the expression a perfect square, you must add 28x^2.
That is not an equation. You mean to add a constant to it? Is that the idea?
9x⁴ + 2x² + 25
= (9x⁴ + 2x² + 1/9) + 25 - 1/9
= (3x² + 1/3)² + 25 - 1/9
= (3x² + 1/3)² + 224/9
So, if you were to add -224/9 to the expression you would have (3x² + 1/3)², a perfect square. Is that what you wanted?
If you add -224/9, then you get 9x^4 + 2x^2 + (1/9) which is equal to (3x^2 + (1/3))^2
There are several things you could add to make it a perfect square. Since you were given a few options, just try each one.
y-25+36/18^2 = 9[x^4+(2/9)x^2+4/18^2]
y-25+1/9 = 9[x^4+(2/9)x^2+1/81]
y - 224/9 = 9(x^2+1/9)^2
y = 9(x^2+1/9)^2 + 224/9
28x^2. Square roots of 9x^4 and 25 are 3x^2 and 5 respectively. A perfect square trinomial can be achieved in one of two ways:
(x - y)^2 = x^2 - 2xy + y^2
(x + y)^2 = x^2 + 2xy + y^2
The only thing that differs is the sign of the middle term. All you do is multiply 3x^2 and 5, then double that product to get 30x^2. You got 2x^2 already, you must add 28x^2 to get the perfect square.
Ans. add in (4x^2-24).
The expression (not equation) is 9x^4 + 2x^2 + 25 or (3x² + 5)² - f(x)
9x⁴ + 2x² + 25 = (3x² + 5)² - f(x)
f(x) = 9x² + 30x² + 25 - 9x⁴ - 2x² - 25 = 28x²
If you add 28x² to the given expression, the result is the perfect square (3x² + 5)²