Question

A6. Подробно, если можно. Спасибо!

Answer 1

$(sqrt{4x+sqrt{16+12x^{2}}})^{2} =(x+2)^{2}\\4x+sqrt{16+12x^{2}}=x^{2}+4x+4\\sqrt{16+12x^{2}}=x^{2}+4\\(sqrt{16+12x^{2}})^{2}=(x^{2}+4)^{2}\\16+12x^{2}=x^{4}+8x^{2}+16\\x^{4}+8x^{2}-12x^{2}=0\\x^{4}-4x^{2}=0$

$x^{2}(x^{2}-4)=0\\x^{2}(x-2)(x+2)=0\\left[begin{array}{ccc}x_{1}=0 \x-2=0\x+2=0end{array}right\\left[begin{array}{ccc}x_{1}=0 \x_{2}=2 \x_{3}=-2 end{array}right$

Проверка:

$1)sqrt{4*0+sqrt{16+12*0^{2}}}=0+2\\sqrt{0+sqrt{16}}=2\\sqrt{4}=2\\2=2-verno\\2)sqrt{4*(-2)+sqrt{16+12*(-2)^{2}}}=-2+2\\sqrt{-8+sqrt{64}}=0\\sqrt{-8+8}=0\\0=0-verno\\3)sqrt{4*2+sqrt{16+12*2^{2}}}=2+2\\sqrt{8+sqrt{64}}=4\\sqrt{8+8}=4\\sqrt{16}=4\\4=4-verno\\Otvet:boxed{0;-2;2}$

Три корня