Find all values of b or c that will make the polynomial a perfect square trinomial. 49x^2+70x+c

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a quadratic expression that can be written in the form (ax + b)², which expands to a²x² + 2abx + b². For a trinomial to be a perfect square, the first term must be a perfect square, the last term must also be a perfect square, and the middle term must be twice the product of the square roots of the first and last terms.

Coefficients in Quadratics

In the polynomial 49x² + 70x + c, the coefficients are the numerical factors of each term. The coefficient of x² is 49, which is a perfect square (7²), and the coefficient of x is 70. Understanding how these coefficients relate to the conditions for forming a perfect square trinomial is essential for determining the value of c.

Finding c for Perfect Square

To find the value of c that makes the polynomial a perfect square trinomial, we can use the relationship between the coefficients. Specifically, we set c equal to the square of half the coefficient of x, which is (70/2)² = 35² = 1225. Thus, c must equal 1225 for the polynomial to be a perfect square trinomial.