How to Solve an Equation Involving a Determinant?

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To solve the equation involving a determinant, the user correctly derived the quadratic equation 2x^2 - 9x - 18 = 0. The next step is to solve this quadratic equation using methods such as factoring, completing the square, or the quadratic formula. After finding the solutions for x, it is essential to verify these solutions by substituting them back into the original determinant equation. This ensures that the solutions are valid within the context of the problem. Properly solving and checking the quadratic will yield the final answers.
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Solve the following equation involving a determinant

det x (x+2)
9 2x = 0
I did x * 2x - 9(x+2)
It came out to 2x^2 - 9x - 18 = 0
Do I have to do anything else like a quadratic equation? I went to a math tutor and he wasn't sure. I scanned the problem and it's number 11.
http://pic20.picturetrail.com/VOL1370/5671323/23539305/392720959.jpg
 
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No. You simply solve the quadratic and check your solutions within the original equation.
 

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Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks
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