if f(x)=ax^2 +bx+c and f(5)=−3f(2) an intersiction point (−4,0) an the X-axis faind another point the X-axis


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Question Number 136667 by mathlove last updated on 24/Mar/21

$i f f (x) = {a x}^{2} + b x + c a n d f (5) = - 3 f (2)$ $a n i n t e r s i c t i o n p o i n t (- 4, 0)$ $a n t h e X - a x i s f a i n d a n o t h e r p o i n t$ $t h e X - a x i s$
	Answered by MJS_new last updated on 25/Mar/21

	$I should have finished it$ $(1) 25 a + 5 b + c = - 3 (4 a + 2 b + c) \Leftrightarrow 37 a + 11 b + 4 c = 0$ $(2) 16 a - 4 b + c = 0$ $(1) \Rightarrow c = - \frac{37 a + 11 b}{4}$ $\Rightarrow (2) \frac{27}{4} a - \frac{27}{4} b = 0 \Rightarrow b = a \Rightarrow c = - 12 a$ $\Rightarrow f (x) = a (x^{2} + x - 12) = a (x - 3) (x + 4)$ $\Rightarrow zeros are - 4 and 3$
	Answered by ajfour last updated on 25/Mar/21
	$25a+5b+c=−3(4a+2b+c) ⇒ 37a+11b+4c=0 (c/a)=−((11)/4)((b/a))−((37)/4) x=−(b/(2a))±((√(b^2 −4ac))/(2a)) x=−(b/(2a))±(√((1/4)((b/a))^2 −(c/a))) x=(b/a){−(1/2)±(√((1/4)+((11)/(4(b/a)))+((37)/(4(b/a)^2 )))) } say (b/a)=p (((−4)/p)+(1/2))^2 =(1/4)+((11)/(4p))+((37)/(4p^2 )) ((16)/p^2 )−(4/p)=((11)/(4p))+((37)/(4p^2 )) ((27)/(4p^2 ))−((27)/(4p))=0 ⇒ p=1 x={−(1/2)±(√((1/4)+((11)/4)+((37)/4))) } x=−(1/2)±(7/2) x=−4 or 3 other point on x-axis is (3,0)$
	$25 a + 5 b + c = - 3 (4 a + 2 b + c)$ $\Rightarrow 37 a + 11 b + 4 c = 0$ $\frac{c}{a} = - \frac{11}{4} (\frac{b}{a}) - \frac{37}{4}$ $x = - \frac{b}{2 a} \pm \frac{\sqrt{b^{2} - 4 a c}}{2 a}$ $x = - \frac{b}{2 a} \pm \sqrt{\frac{1}{4} {(\frac{b}{a})}^{2} - \frac{c}{a}}$ $x = \frac{b}{a} {- \frac{1}{2} \pm \sqrt{\frac{1}{4} + \frac{11}{4 (b / a)} + \frac{37}{4 {(b / a)}^{2}}}}$ $s a y \frac{b}{a} = p$ ${(\frac{- 4}{p} + \frac{1}{2})}^{2} = \frac{1}{4} + \frac{11}{4 p} + \frac{37}{4 p^{2}}$ $\frac{16}{p^{2}} - \frac{4}{p} = \frac{11}{4 p} + \frac{37}{4 p^{2}}$ $\frac{27}{4 p^{2}} - \frac{27}{4 p} = 0$ $\Rightarrow p = 1$ $x = {- \frac{1}{2} \pm \sqrt{\frac{1}{4} + \frac{11}{4} + \frac{37}{4}}}$ $x = - \frac{1}{2} \pm \frac{7}{2}$ $x = - 4 o r 3$ $o t h e r p o i n t o n x - a x i s i s$ $(3, 0)$
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