Question

Giải các phương trình :

a) \(\sqrt{5x+3}=3x-7\)

b) \(\sqrt{3x^2-2x-1}=3x+1\)

c) \(\dfrac{\sqrt{4x^2+7x-2}}{x+2}=\sqrt{2}\)

d) \(\sqrt{2x^2+3x-4}=\sqrt{7x+2}\)

Answer 1

a) \(\sqrt{5x+3}=3x-7\)\(\Leftrightarrow\left\{{}\begin{matrix}5x+3=\left(3x-7\right)^2\\3x-7\ge0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}5x+3=9x^2-42x+49\\x\ge\dfrac{7}{3}\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}9x^2-47x+46=0\\x\ge\dfrac{7}{3}\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\left[{}\begin{matrix}x=\dfrac{47+\sqrt{553}}{18}\\x=\dfrac{47-\sqrt{553}}{18}\end{matrix}\right.\\x\ge\dfrac{7}{3}\end{matrix}\right.\)\(\Leftrightarrow\dfrac{47+\sqrt{553}}{18}\).

Answer 2

b) \(\sqrt{3x^2-2x-1}=3x+1\)\(\Leftrightarrow\left\{{}\begin{matrix}3x^2-2x-1=\left(3x+1\right)^2\\3x+1\ge0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}6x^2+8x+2=0\\x\ge\dfrac{-1}{3}\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}\left[{}\begin{matrix}x=-\dfrac{1}{3}\\x=-1\end{matrix}\right.\\x\ge-\dfrac{1}{3}\end{matrix}\right.\)\(\Leftrightarrow x=-\dfrac{1}{3}\).

Answer 3

c) \(\dfrac{\sqrt{4x^2+7x-2}}{x+2}=\sqrt{2}\)\(\Rightarrow\sqrt{4x^2+7x-2}=\sqrt{2}\left(x+2\right)\)
\(\Leftrightarrow\left\{{}\begin{matrix}4x^2+7x-2=\left[\sqrt{2}\left(x+2\right)\right]^2\\x+2\ge0\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}2x^2-15x-10=0\\x\ge-2\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}\left[{}\begin{matrix}x=-2\\x=\dfrac{5}{2}\end{matrix}\right.\\x\ge-2\end{matrix}\right.\)
​Thử lại các nghiệm vào phương trình ta thấy \(x=\dfrac{5}{2}\) là nghiệm của phương trình.

Answer 4

d) \(\sqrt{2x^2+3x-4}=\sqrt{7x+2}\)\(\Leftrightarrow\left\{{}\begin{matrix}2x^2+3x-4=7x+2\\7x+2\ge0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}2x^2-4x-6=0\\x\ge\dfrac{-2}{7}\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}\left[{}\begin{matrix}x=-1\\x=3\end{matrix}\right.\\x\ge\dfrac{-2}{7}\end{matrix}\right.\)\(\Leftrightarrow x=3\).