To factor the expression \(100x^2 + 20x + 1\), we can look for two numbers that multiply to give the product of the leading coefficient (100) and the constant term (1), which is \(100 \cdot 1 = 100\), and also add up to the middle coefficient (20).
The expression can be factored as follows:
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Identify the coefficients:
- \(a = 100\)
- \(b = 20\)
- \(c = 1\)
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Calculate the product:
- \(a \cdot c = 100 \cdot 1 = 100\)
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Find two numbers that multiply to 100 and add to 20:
- The numbers are \(10\) and \(10\) because \(10 \cdot 10 = 100\) and \(10 + 10 = 20\).
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Rewrite the quadratic: \[ 100x^2 + 10x + 10x + 1 \]
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Group the terms: \[ (100x^2 + 10x) + (10x + 1) \]
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Factor by grouping: \[ 10x(10x + 1) + 1(10x + 1) \]
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Factor out the common binomial factor: \[ (10x + 1)(10x + 1) \text{ or } (10x + 1)^2 \]
Therefore, the completely factored form of \(100x^2 + 20x + 1\) is: \[ (10x + 1)^2 \]
So, the final answer is: \[ 100x^2 + 20x + 1 = (10x + 1)^2 \]