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Answer
\( 7d^{2} + 9d + 2 = (d + 1)(7d + 2) \)
Solution
To factor the quadratic expression 7d² + 9d + 2, follow these steps:
Step 1: Multiply the coefficient of d² (which is 7) by the constant term (which is 2), giving 7 × 2 = 14.
Step 2: Find two numbers that multiply to 14 and add to the coefficient of d (which is 9). The numbers 7 and 2 work since 7 × 2 = 14 and 7 + 2 = 9.
Step 3: Rewrite the middle term 9d as 7d + 2d:
7d² + 7d + 2d + 2
Step 4: Factor by grouping:
Group the first two terms and the last two terms:
(7d² + 7d) + (2d + 2)
Factor out the common factors in each group:
7d(d + 1) + 2(d + 1)
Step 5: Notice that (d + 1) is common to both groups. Factor (d + 1) out:
(d + 1)(7d + 2)
Thus, the completely factored form of 7d² + 9d + 2 is (d + 1)(7d + 2).
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Simplify this solution Mind Expander
To factor the quadratic expression \( 7d^{2} + 9d + 2 \) completely, we look for two numbers that multiply to \( 7 \times 2 = 14 \) and add up to \( 9 \). Those numbers are \( 7 \) and \( 2 \). We can then rewrite the middle term of the equation: \( 7d^{2} + 7d + 2d + 2 \) Next, we can group the terms: \( (7d^{2} + 7d) + (2d + 2) \) Factoring out the common factors in each group gives us: \( 7d(d + 1) + 2(d + 1) \) Now, we can factor out the common factor \( (d + 1) \): \( (7d + 2)(d + 1) \) So, the completely factored form of \( 7d^{2} + 9d + 2 \) is \( (7d + 2)(d + 1) \).
