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Help algebra

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The teacher has written an equation of the form  on the board, where  is a quadratic, but Heather can't read the linear term. She can see that the quadratic term is  4x^2 and that the constant is -24. She asks her neighbor, Noel, what the linear term is.

Noel decides to tease her and just says, One of the roots is 3.''

Heather then says, Oh, thanks!''

She then correctly writes down the linear term. What was the linear term? Include entire term, not simply the coefficient.

May 31, 2021

#1
+26213
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The teacher has written an equation of the form  on the board,
She can see that the quadratic term is  4x^2 and that the constant is -24.
She asks her neighbor, Noel, what the linear term is.
Noel decides to tease her and just says, One of the roots is 3.''

Heather then says, Oh, thanks!''

She then correctly writes down the linear term.
What was the linear term? Include entire term, not simply the coefficient.

$$\begin{array}{|rcll|} \hline 4x^2+ax-24&=& 0 \\ \hline \dfrac{-24}{4} &=& r_1*r_2 \quad | \quad r_1 = 3 \\\\ \dfrac{-24}{4} &=& 3*r_2 \\\\ r_2 &=& \dfrac{-24}{12} \\\\ \mathbf{r_2} &=& \mathbf{-2} \\ \hline \dfrac{a}{4} &=& - (r_1+r_2) \\\\ \dfrac{a}{4} &=& - (3-2) \\\\ \mathbf{ a } &=& \mathbf{ -4 } \\ \hline \end{array}$$

$$4x^2-4x-24$$

May 31, 2021