\(3\left(4x-5\right)^2-8x+10=0\)

\(\Leftrightarrow\)\(3\left(4x-5\right)\left(4x-5\right)-2\left(4x-5\right)=0\)

\(\Leftrightarrow\)\(\left(4x-5\right)\left[2\left(4x-5\right)-2\right]=0\)

\(\Leftrightarrow\)\(\left(4x-5\right)\left(8x-10-2\right)=0\)

\(\Leftrightarrow\)\(\left(4x-5\right)\left(8x-12\right)=0\)

\(\Leftrightarrow\)\(\orbr{\begin{cases}4x-5=0\\8x-12=0\end{cases}}\)

\(\Leftrightarrow\)\(\orbr{\begin{cases}x=\frac{5}{4}\\x=1,5\end{cases}}\)

Vậy...

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\(3\left(4x-5\right)^2-8x+10=0\)

<=> \(3\left(16x^2-40x+25\right)-8x+10=0\)

<=> \(48x^2-120x+75-8x+10=0\)

<=> \(48x^2-128x+85=0\)

<=> \(48x^2-68x-60x-85=0\)

<=> \(48x\left(x-\frac{17}{12}\right)-60\left(x-\frac{17}{12}\right)=0\)

<=> \(\left(48x-60\right)\left(x-\frac{17}{12}\right)=0\)

<=> \(\hept{\begin{cases}48x-60=0\\x-\frac{17}{12}=0\end{cases}}\)

<=> \(\hept{\begin{cases}x=\frac{5}{4}\\x=\frac{17}{12}\end{cases}}\)

\(A=16x^2-y^2-16x^2+8x=8x-y^2\\ A=8\cdot3-\left(-1\right)^2=24-1=23\\ B=64x^3-80x-64x^3-1=-80x-1\\ B=-80\cdot\dfrac{1}{5}-1=-16-1=-17\)

\(A=\sqrt{\left(x+2\right)^2+7}+\sqrt{\left(x-4\right)^2+7}\)

Dạng bài này sử dụng bất đẳng thức Mincopxki \(\sqrt{a^2+b^2}+\sqrt{c^2+d^2}\ge\sqrt{\left(a+c\right)^2+\left(b+d\right)^2}\text{ }\left(1\right)\)

Chứng minh: 

\(\left(1\right)\Leftrightarrow a^2+b^2+c^2+d^2+2\sqrt{a^2+b^2}.\sqrt{c^2+d^2}\ge\left(a+c\right)^2+\left(b+d\right)^2\)

\(\Leftrightarrow\sqrt{\left(a^2+b^2\right)\left(c^2+d^2\right)}\ge ac+bd\)

\(+\text{Nếu }ac+bd< 0\text{ thì }VT\ge0>VP,\text{ bđt luôn đúng.}\)

\(\text{+Nếu }ac+bd>0\)

\(\text{bđt}\Leftrightarrow\left(a^2+b^2\right)\left(c^2+d^2\right)\ge\left(ac+bd\right)^2\)

\(\Leftrightarrow\left(ad-bc\right)^2\ge0\)

Do bđt cuối đúng nên bất đẳng thức đã cho cũng đúng.

Vậy ta có đpcm.

Dấu bằng xảy ra khi \(ad=bc\)

\(A=\sqrt{\left(x+2\right)^2+\left(\sqrt{7}\right)^2}+\sqrt{\left(4-x\right)^2+\left(\sqrt{7}\right)^2}\)

\(\ge\sqrt{\left(x+2+4-x\right)^2+\left(\sqrt{7}+\sqrt{7}\right)^2}\)

\(=\sqrt{64}=8.\)

Dấu bằng xảy ra khi \(\left(x+2\right).\sqrt{7}=\left(4-x\right).\sqrt{7}\Leftrightarrow x+2=4-x\Leftrightarrow x=1.\)

Vậy GTNN của biểu thức là 8.