Question

Find the value of p if the equation 3x²- 2x+p=0 and 6x²-17x+12=0 have a common root

Answer 1

Question :-

Find the value of p if the equation 3x²- 2x+p=0 and 6x²-17x+12=0 have a common root

Solution :-

Here, we have two quadratic equations and also said they have same roots. And we need to find the value of p is in the eqⁿ 1. Then we find the roots of 2nd equation and then find the value of p.

By splitting the middle term,

$\sf:\longmapsto{6x {}^{2} - 8x - 9x + 12 = 0} \: \: \: \: \: \: \\ \\ \sf:\longmapsto{2x (3x - 4) - 3(3x - 4) = 0 } \\ \\ \sf:\longmapsto{(3x - 4)(2x - 3) = 0} \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf:\longmapsto{3x - 4 = 0} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \sf:\longmapsto{x = \frac{4}{3} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf:\longmapsto{2x - 3 = 0} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \sf:\longmapsto{x = \frac{3}{2} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

we have roots :

• x = 4/3
• x = 3/2

Now, by substituting method find the value of p.

$\sf:\longmapsto{3 \bigg( \frac{4}{3} \bigg) {}^{2} - 2 \bigg( \frac{4}{3} \bigg) + p = 0 } \\ \\ \sf:\longmapsto{3 \times \frac{16}{9} - \frac{8}{3} + p = 0 } \: \: \: \: \: \: \: \: \: \\ \\ \sf:\longmapsto{ \frac{16}{3} - \frac{8}{3} + p = 0 } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\\sf:\longmapsto{ \frac{16 - 8}{3} + p = 0 } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf:\longmapsto{ \frac{8}{3} + p = 0 } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \bf:\longmapsto \red{p = - \frac{8}{ 3} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

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$\sf:\longmapsto{3 \bigg( \frac{3}{2} \bigg) {}^{2} - 2 \times \frac{3}{2} + p = 0} \\ \\ \sf:\longmapsto{3 \times \frac{9}{4} - 3 + p = 0 } \: \: \: \: \: \: \: \: \: \: \\ \\ \sf:\longmapsto{ \frac{27}{4} - 3 } + p = 0 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf:\longmapsto{ \frac{27 - 12}{4} + p = 0 } \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf:\longmapsto{ \frac{15}{4} } + p = 0 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \bf:\longmapsto \red{p = - \frac{15}{4} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

The values of p are $\rm{ - \frac{15}{4} , - \frac{8}{3} }$.

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Learn more from brainly :

Is (3x+2) a factor of p(x) = x²+x².17x+15

sol 3x+2=0.

https://brainly.in/question/42140316