Verified Answer :

  • Identify coefficients \(A=2\), \(B=7\), and \(C=5\).
  • Find two numbers \(p=2\) and \(q=5\) that satisfy \(p \times q = 10\) and \(p + q = 7\).
  • Rewrite the expression as \(2t^2 + 2t + 5t + 5\).
  • Factor by grouping to obtain the result \((2t + 5)(t + 1)\).
  • Final factored form: \(\boxed{(2t+5)(t+1)}\).

Explanation

  1. Analyze the Quadratic Trinomial Coefficients

To factor the quadratic trinomial \(2t^2 + 7t + 5\), we identify its coefficients as \(A=2\), \(B=7\), and \(C=5\). The standard method for factoring a trinomial of the form \(At^2 + Bt + C\) is to look for two numbers, \(p\) and \(q\), whose product is equal to \(A \times C\) and whose sum is equal to \(B\). Here, \(A \times C = 2 \times 5 = 10\) and \(B = 7\).

  1. Find Two Numbers with Specific Product and Sum

We need to find two numbers \(p\) and \(q\) such that:
\[p \times q = 10\]
\[p + q = 7\]
By inspecting the factors of 10, we observe that the numbers 2 and 5 satisfy these conditions, as \(2 \times 5 = 10\) and \(2 + 5 = 7\).

  1. Factor by Grouping

Now, we rewrite the middle term \(7t\) using these two numbers:
\[2t^2 + 2t + 5t + 5\]
This allows us to group the terms:
\[(2t^2 + 2t) + (5t + 5)\]
Next, factor out the common term from each group:
\[2t(t + 1) + 5(t + 1)\]
Finally, factor out the common binomial \((t + 1)\):
\[(2t + 5)(t + 1)\]
Thus, the factored form of the expression is \((2t + 5)(t + 1)\).

Examples

Factoring quadratic expressions is a fundamental skill used in physics to determine when a projectile hits the ground. For instance, if the height of an object thrown upward is given by a quadratic expression like \(h(t) = -5t^2 + 20t + 25\), you can factor out common terms to solve for \(h(t) = 0\). This helps in calculating the exact time at which an object returns to the ground, which is crucial for engineering and motion analysis. Mastering these algebraic techniques provides the foundation for solving more complex real-world dynamics problems.

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