Question

Write explanation to the right of each step you would explain the following algebraic problem to learner
2.3*=162
3*=81
3*=34
X=4

Fractionize and solve the unknown in x square + 3x + 2

Answer 1

We should solve an equation

$2\cdot 3^x = 162.$ First, we should divide each hand of the equation by 2, it helps us to obtain only the exponential term on the left hand of the equation.

$3^x=81.$ Next , we should determine, in which degree 3 becomes equal to 81. We know that $3^1=3, 3^2=9, 3^3=27, 3^4=81.$

$3^x=3^4.$ We know that the exponential function is monotonic, so this equation has a single root, $x=4.$

We should factorize $x^2+3x+2$ . That means, we should transform it into $(x-x_1)(x-x_2).$ If we expand this, we'll get $x^2-xx_1-xx_2+x_1x_2 = x^2-x(x_1+x_2)+x_1x_2.$

Therefore, $x_1x_2 = 2, \;\; x_1+x_2 = -3.$ We may find such values, $x_1=-1, x_2 = -2.$

Therefore, the factorization takes form $(x-(-1))(x-(-2)) = (x+1)(x+2).$