Which equation has the solutions mc013-1.jpg?

Asked by maham237 @ 26/12/2020 in Mathematics viewed by 498 People

Which equation has the solutions mc013-1.jpg? 3x2 – 5x + 7 = 0 3x2 – 5x – 1 = 0 3x2 – 10x + 6 = 0 3x2 – 10x – 1 = 0

Answered by maham237 @ 26/12/2020

Given\:solutions: $\frac{5+2\sqrt{7}}{3}\:and\:\frac{5-2\sqrt{7}}{3}$ .

Therefore, factors of the equation would be $(x-\frac{5+2\sqrt{7}}{3})\:and\:(x-\frac{5-2\sqrt{7}}{3})$

Let us multiply those two factors to get the equation, we get

$\left(x-\frac{5+2\sqrt{7}}{3}\right)\left(x-\frac{5-2\sqrt{7}}{3}\right)$

$\mathrm{Apply\:FOIL\:method}:\quad \left(a+b\right)\left(c+d\right)=ac+ad+bc+bd$

$a=x,\:b=-\frac{5+2\sqrt{7}}{3},\:c=x,\:d=-\frac{5-2\sqrt{7}}{3}$

$=xx+x\left(-\frac{5-2\sqrt{7}}{3}\right)+\left(-\frac{5+2\sqrt{7}}{3}\right)x+\left(-\frac{5+2\sqrt{7}}{3}\right)\left(-\frac{5-2\sqrt{7}}{3}\right)$

$=xx-\frac{5-2\sqrt{7}}{3}x-\frac{5+2\sqrt{7}}{3}x+\frac{5+2\sqrt{7}}{3}\cdot \frac{5-2\sqrt{7}}{3}$

$=x^2-\frac{5x-2\sqrt{7}x}{3}-\frac{5x+2\sqrt{7}x}{3}-\frac{1}{3}$

$\mathrm{Combine\:the\:fractions\:}-\frac{5x-2\sqrt{7}x}{3}-\frac{5x+2\sqrt{7}x}{3}-\frac{1}{3}:\quad \frac{-\left(5x-2\sqrt{7}x\right)-\left(5x+2\sqrt{7}x\right)-1}{3}$

$=x^2+\frac{-\left(5x-2\sqrt{7}x\right)-\left(5x+2\sqrt{7}x\right)-1}{3}$

$\mathrm{Expand}\:-\left(5x-2\sqrt{7}x\right)-\left(5x+2\sqrt{7}x\right)-1:\quad -10x-1$

$=x^2+\frac{-10x-1}{3}$

Setting it equal to 0.

$x^2+\frac{-10x-1}{3}=0$

Multiplying whole equation by 3, we get

$3x^2-10x-1 = 0$

Therefore, correct option is D $3x^2-10x-1 = 0$

Which equation has the solutions mc013-1.jpg?

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