Question

Find the term that should be added to the expression to create a perfect square trinomial. $$x^{2}-10 x$$

Short answer

The term that should be added is 25.

Step-by-step solution

Understand the problem.

We are given a polynomial \(x^{2}-10 x\). We want to convert this polynomial into a perfect square trinomial. A perfect square trinomial is of the form \(a^{2}+2ab+b^{2}\) or \(a^{2}-2ab+b^{2}\). Here, a = x. So let’s find the value of b.

Formula to find 'b' of perfect square trinomial.

The perfect square trinomial formula when a = x is \(x^{2}+2bx+b^{2}\) for positive trinomial and \(x^{2}-2bx+b^{2}\) for negative trinomial. Given \(x^{2}-10 x\), this resembles the structure of the negative perfect square trinomial but it's missing \(b^{2}\).

Applying the formula.

We know that in the given expression, -10x is equal to -2bx (where x is 'a' and 'b' is what we need to find). Hence, we equate -2bx to -10x to get the value of b. We get \(b = \frac{-10x}{-2x} = 5\). Hence, the missing term is \(b^{2} = 5^{2} = 25\).

Conclusion.

The term that should be added to the expression \(x^{2} - 10x\) to create a perfect square trinomial is \(25\).