Complete the expression blank minus 60 𝑥 squared plus 25 to make a perfect square.
If we’re only trying to form a perfect square, the first and the last terms must be positive perfect squares, and the middle term must be twice the product of their square roots. And since we know the middle term is negative 60 𝑥 squared, we can use that and then find the first term, because we also know the third term, the last term.
So we will replace the word “middle term” with negative 60 𝑥 squared. Now notice that we have a plus and a minus next to the two on the other side of the equation. This deciphers the sign of the middle term. Well we already know the sign of the middle term. It’s negative. So we need to use a negative two.
And since we don’t know the first term, it’s ok to just leave it like this because we will be solving for it. And then the last term, the third term, is 25. So we can go ahead and simplify the square root of 25 to be five. So on the right-hand side of the equation, we have a negative two and we have a five. Let’s go ahead and multiply those together.
So now we need to solve for that first term. And to do so, we need to divide both sides of the equation by negative 10. It cancels on the right-hand side. And negative 60 divided by negative 10 is equal to six. So we have six 𝑥 squared is equal to the square root of the first term. So in order to solve for the first term, we need to square both sides of the equation.
And when doing so, we square six to get 36 and we square the 𝑥 squared to get 𝑥 to the fourth. Therefore, the first term that we need in order to make this expression a perfect square would be 36𝑥 to the fourth power.