Given that one root is 3 times the others for the quadratic equation $3 x^{2} - 2 x + p = 0$ $3x^2-2x+p=0$ , find (a) the value of $p$ $p$ and (b) the two roots ?

Question

Given that one root is 3 times the others for the quadratic equation $3 x^{2} - 2 x + p = 0$ $3x^2-2x+p=0$ , find (a) the value of $p$ $p$ and (b) the two roots ?

1 Answer

Impact of this question

16007 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-05-04T13:30:48

$p = \frac{1}{4}, x = \frac{1}{6}, \frac{1}{2}$ $p=1/4, x=1/6, 1/2$

Explanation:

Let the roots be $r, 3 r$ $r, 3r$ :

$a (x - r) (x - 3 r) = 0$ $a(x-r)(x-3r)=0$

$a x^{2} - 4 a r x + 3 a r^{2} = 0$ $ax^2-4arx+3ar^2=0$

$3 x^{2} - 2 x + p = a x^{2} - 4 a r x + 3 a r^{2}$ $3x^2-2x+p=ax^2-4arx+3ar^2$

$a = 3$ $a=3$

$3 x^{2} - 2 x + p = 3 x^{2} - 12 r x + 9 r^{2}$ $3x^2-2x+p=3x^2-12rx+9r^2$

$12 r = 2$ $12r=2$

$r = \frac{1}{6} \Rightarrow$ $r=1/6=>$ $3 r = \frac{1}{2}$ $3r=1/2$

$p = 9 \cdot \frac{1}{36} = \frac{1}{4}$ $p=9*1/(36)=1/4$

Check:

$3 x^{2} - 2 x + \frac{1}{4} = 0$ $3x^2-2x+1/4=0$
$3 (x^{2} - (\frac{2}{3}) x) = - \frac{1}{4}$ $3(x^2-(2/3)x)=-1/4$
$3 (x^{2} (- \frac{2}{3}) x + {(\frac{1}{3})}^{2}) = \frac{1}{3} - \frac{1}{4} = \frac{1}{12}$ $3(x^2(-2/3)x+(1/3)^2)=1/3-1/4=1/12$
${(x - \frac{1}{3})}^{2} = \frac{1}{36}$ $(x-1/3)^2=1/36$
$x - \frac{1}{3} = \pm \frac{1}{6}$ $x-1/3=+-1/6$
$x = \frac{1}{3} \pm \frac{1}{6}$ $x=1/3+-1/6$
$x = \frac{1}{3} - \frac{1}{6} = \frac{2}{6} - \frac{1}{6} = \frac{1}{6}$ $x=1/3-1/6=2/6-1/6=1/6$
$x = \frac{1}{3} + \frac{1}{6} = \frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}$ $x=1/3+1/6=2/6+1/6=3/6=1/2$

Science

Math

Humanities

... and beyond

Given that one root is 3 times the others for the quadratic equation $3 x^{2} - 2 x + p = 0$ $3x^2-2x+p=0$ , find (a) the value of $p$ $p$ and (b) the two roots ?

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

Given that one root is 3 times the others for the quadratic equation 3x^2-2x+p=03x2−2x+p=03x^2-2x+p=0, find (a) the value of ppp and (b) the two roots ?

1 Answer

Explanation:

Related questions

Impact of this question

Given that one root is 3 times the others for the quadratic equation $3 x^{2} - 2 x + p = 0$ $3x^2-2x+p=0$ , find (a) the value of $p$ $p$ and (b) the two roots ?