Question

Find the roots of the equation (x² + 3x)² – (x² + 3x) –6 = 0.

please solve with explaination about why we should take x²+3x as y​

Answer 1

Answer:

(- 3 + √21) / 2, (- 3 - √21) / 2, - 1, - 2

Step-by-step explanation:

Of course we can open parentheses and solve 4th degree polynomial equation. But it will be long way, which will, any way, lid us to method of substitution.

The shorter and rational way is to substitute (x² + 3x) by "y" and first solve the quadratic equation.

So, now the given equation will be

y² - y - 6 = 0

$y_{1}$ + $y_{2}$ = 1

$y_{1}$ × $y_{2}$ = - 6

$y_{1}$ = 3

$y_{2}$ = - 2

Now let's find value of "x"

x² + 3x = 3 ===> x² + 3x - 3 = 0

$x_{1}$ = (- 3 + √21) / 2

$x_{2}$ = (- 3 - √21) / 2

x² + 3x = - 2 ===> x² + 3x + 2 = 0

$x_{3}$ = - 1

$x_{4}$ = - 2