To complete the polynomial equation, we need to subtract the second polynomial from the first and find the missing values. The equation starts as follows:
\[ 9x^2 + ___x + 13
- 2x^2 + 12x + ___ \]
Now, let's perform the subtraction step by step.
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Subtract the coefficients of \(x^2\): \[ 9x^2 - 2x^2 = 7x^2 \]
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Subtract the coefficients of \(x\): We want to find the coefficient of \(x\) in the first polynomial to subtract 12 from it. Let’s denote it as \(a\): \[ a - 12 = 5 \] Solving for \(a\): \[ a = 5 + 12 = 17 \]
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Subtract the constant terms: We need to find the missing constant in the second polynomial. Let’s denote it as \(b\): \[ 13 - b = 2 \] Solving for \(b\): \[ b = 13 - 2 = 11 \]
Now we have all of the values:
- The coefficient of \(x\) in the first polynomial is \(17\).
- The missing constant from the second polynomial is \(11\).
Bringing it all together, we can fill in the blanks:
\[ 9x^2 + 17x + 13
- 2x^2 + 12x + 11 = 7x^2 + 5x + 2 \]
So, the completed equation is:
\[ \begin{aligned} &9x^2 + 17x + 13 \ &- 2x^2 + 12x + 11 \ &= 7x^2 + 5x + 2 \end{aligned} \]
Final Answer: 17 for the first blank (coefficient of \(x\)), 11 for the second blank (constant in the second polynomial).
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