Question

Qual é o valor de x?

Por favor peço que me expliquem e não apenas a resposta okay? Thank you so much!

(x+9)²=(x+3)²+2x²

Answer 1

Given (x + 9)² = (x + 3)² + 2x², solve for x.

(x + 9)² = (x + 3)² + 2x²

Developing the polynomials

x² + 18x + 81 = x² + 6x + 9 + 2x²

Isolating all terms in the RHS

0 = x² + 6x + 9 + 2x² - x² - 18x - 81

Inverting

x² + 6x + 9 + 2x² - x² - 18x - 81 = 0

Combining the like terms

2x² - 12x - 72 = 0

Dividing both sides by 2

x² - 6x - 36 = 0

Applying the quadratic formula (Bhaskara) when the coefficient of x is even, that is to say, "b" in Bhaskara, is even.

$ax^{2} +bx+c=0$

Where a ≠ 0, and b = 2n, for some integer n

Take a look at the formula:

$x=\frac{-b'+\sqrt{b'^{2}-ac}}{a}, \frac{-b'-\sqrt{b'^{2}-ac}}{a}$

Where b' is the half of b

Hence, applying the formula in "x² - 6x - 36 = 0"

$x^{2} -6x-36\\x=\frac{3+-\sqrt{3^{2}-(-36)}}{1}\\x=3+-\sqrt{9+36}\\x=3+-\sqrt{45}\\\\\left \{ {{x=3+3\sqrt{5}} \atop {x=3-3\sqrt{5}}} \right.$

Thus, the solutions are:

x = 3 ± 3√5

Hope I've helped, enjoy your studies.