Find all values of b or c that will make the polynomial a perfect square trinomial. 100r²-60r+c

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Recognize that a perfect square trinomial takes the form $\left(\sqrt{A}r + d\right)^2 = Ar^2 + 2d\sqrt{A}r + d^2$, where $A$ is the coefficient of $r^2$ and $d$ is a constant to be determined.

Identify the coefficient of $r^2$ in the given polynomial: \$100r^2$. So, $A = 100$ and $\sqrt{A} = 10$.

Compare the middle term of the polynomial, $-60r$, to the middle term of the perfect square form, $2d \times 10r = 20d r$. Set $20d = -60$ to solve for $d$.

Solve for $d$ by dividing both sides of the equation $20d = -60$ by 20, which gives $d = -3$.

Find the constant term $c$ by squaring $d$: $c = d^2 = (-3)^2$. This value of $c$ will make the polynomial a perfect square trinomial.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Perfect Square Trinomial

A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial, typically in the form (ax + b)^2 = a^2x^2 + 2abx + b^2. Recognizing this form helps identify the necessary constant term to complete the square.

Identifying Coefficients in Quadratics

In a quadratic expression ax^2 + bx + c, the coefficients a, b, and c determine its shape and factorization. Understanding how these coefficients relate, especially the middle term and constant, is essential to determine if the polynomial is a perfect square.

Completing the Square

Completing the square involves rewriting a quadratic expression so that it forms a perfect square trinomial. This process often requires finding a specific constant term c that makes the expression factorable as (mx + n)^2, which is key to solving the given problem.

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