25r²-30r+9 square root of first term

Mathematics
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25r²-30r+9 square root of first term

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25r²-30r+9 square root of first term

Answer:
Let’s solve the given mathematical expression step-by-step. The expression we have is:

25r^2 - 30r + 9

To make sure we thoroughly understand the concept, we’ll break the expression down into simpler parts and then tackle the specific part where we need to find the square root of the first term. Let’s cover these steps in detail:

Step 1: Understanding the Expression

  1. Identify the Terms:

    • The expression 25r^2 - 30r + 9 is a quadratic trinomial (a polynomial with three terms, where the highest power of the variable is 2).

  2. Identify the Standard Form:

    • The standard form of a quadratic expression is ax^2 + bx + c . In our expression:
      • ( a = 25 )
      • ( b = -30 )
      • ( c = 9 )

Step 2: Factoring the Quadratic Expression

Let’s see if the quadratic trinomial can be factored. This step can also help us verify any properties we might use later.

  1. Determine if it’s a Perfect Square Trinomial:

    • A perfect square trinomial is of the form ( (mx + n)^2 ).
    • To check, we can use the formula for expanding a square of a binomial:
      (mx + n)^2 = m^2x^2 + 2mnx + n^2

  2. Compare the Terms:

    • We need to see if 25r^2 - 30r + 9 matches m^2r^2 + 2mnx + n^2

    • Check:

      • m^2r^2 \rightarrow 25r^2 \rightarrow m = 5
      • n^2 \rightarrow 9 \rightarrow n = 3
      • 2mn \rightarrow 2 * 5 * 3 = 30 \rightarrow -30r (the middle term should be negative)

We can confirm our expression is a perfect square trinomial since:
(5r - 3) ^ 2 = 25r^2 - 30r + 9

Step 3: Square Root of the First Term

To find the square root of the first term of our quadratic expression 25r^2

  1. Identify the First Term:

    • The first term in the expression is 25r^2

  2. Taking the Square Root:

    • Use the property of square roots which states ( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} ).
    • Apply this property to ( 25r^2 ):
      \sqrt{25r^2} = \sqrt{25} \times \sqrt{r^2}
      • \sqrt{25} = 5 (since 5^2 = 25
      • \sqrt{r^2} = r (since r^2 is the square of r

Final Result

\sqrt{25r^2} = 5r

Therefore, the square root of the first term 25r^2 is 5r.

If you have any further questions or need additional explanations, feel free to ask!