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Question Number 72782 by Mr. K last updated on 02/Nov/19

Commented by Mr. K last updated on 02/Nov/19

Find a, b and c

$${Find}\:{a},\:{b}\:{and}\:{c} \\ $$

Answered by MJS last updated on 02/Nov/19

y=ax^2 +bx+c  x=0∧y=−4 ⇒ (1)  c=−4  x=2∧y=−4 ⇒ (2)  4a+2b+c=−4  x=6∧y=8 ⇒ (3)  36a+6b+c=8    (2)  4a+2b=0 ⇒ b=−2a  (3)  36a−12a−4=8 ⇒ a=(1/2) ⇒ b=−1  y=(1/2)x^2 −x−4

$${y}={ax}^{\mathrm{2}} +{bx}+{c} \\ $$$${x}=\mathrm{0}\wedge{y}=−\mathrm{4}\:\Rightarrow\:\left(\mathrm{1}\right)\:\:{c}=−\mathrm{4} \\ $$$${x}=\mathrm{2}\wedge{y}=−\mathrm{4}\:\Rightarrow\:\left(\mathrm{2}\right)\:\:\mathrm{4}{a}+\mathrm{2}{b}+{c}=−\mathrm{4} \\ $$$${x}=\mathrm{6}\wedge{y}=\mathrm{8}\:\Rightarrow\:\left(\mathrm{3}\right)\:\:\mathrm{36}{a}+\mathrm{6}{b}+{c}=\mathrm{8} \\ $$$$ \\ $$$$\left(\mathrm{2}\right)\:\:\mathrm{4}{a}+\mathrm{2}{b}=\mathrm{0}\:\Rightarrow\:{b}=−\mathrm{2}{a} \\ $$$$\left(\mathrm{3}\right)\:\:\mathrm{36}{a}−\mathrm{12}{a}−\mathrm{4}=\mathrm{8}\:\Rightarrow\:{a}=\frac{\mathrm{1}}{\mathrm{2}}\:\Rightarrow\:{b}=−\mathrm{1} \\ $$$${y}=\frac{\mathrm{1}}{\mathrm{2}}{x}^{\mathrm{2}} −{x}−\mathrm{4} \\ $$

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