Solve for a (complex solution)
\left\{\begin{matrix}a=\frac{y-1}{\left(x-3\right)^{2}}\text{, }&x\neq 3\\a\in \mathrm{C}\text{, }&y=1\text{ and }x=3\end{matrix}\right.
Solve for a
\left\{\begin{matrix}a=\frac{y-1}{\left(x-3\right)^{2}}\text{, }&x\neq 3\\a\in \mathrm{R}\text{, }&y=1\text{ and }x=3\end{matrix}\right.
Solve for x (complex solution)
\left\{\begin{matrix}x=3+a^{-\frac{1}{2}}\sqrt{y-1}\text{; }x=3-a^{-\frac{1}{2}}\sqrt{y-1}\text{, }&a\neq 0\\x\in \mathrm{C}\text{, }&y=1\text{ and }a=0\end{matrix}\right.
Solve for x
\left\{\begin{matrix}x=-\sqrt{-\frac{1-y}{a}}+3\text{; }x=\sqrt{-\frac{1-y}{a}}+3\text{, }&y\leq 1\text{ and }a<0\\x=\frac{-\sqrt{y-1}+3\sqrt{a}}{\sqrt{a}}\text{; }x=\frac{\sqrt{y-1}+3\sqrt{a}}{\sqrt{a}}\text{, }&y\geq 1\text{ and }a>0\\x\in \mathrm{R}\text{, }&y=1\text{ and }a=0\end{matrix}\right.
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y-1=a\left(x^{2}-6x+9\right)
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.
y-1=ax^{2}-6ax+9a
Use the distributive property to multiply a by x^{2}-6x+9.
ax^{2}-6ax+9a=y-1
Swap sides so that all variable terms are on the left hand side.
\left(x^{2}-6x+9\right)a=y-1
Combine all terms containing a.
\frac{\left(x^{2}-6x+9\right)a}{x^{2}-6x+9}=\frac{y-1}{x^{2}-6x+9}
Divide both sides by x^{2}-6x+9.
a=\frac{y-1}{x^{2}-6x+9}
Dividing by x^{2}-6x+9 undoes the multiplication by x^{2}-6x+9.
a=\frac{y-1}{\left(x-3\right)^{2}}
Divide y-1 by x^{2}-6x+9.
y-1=a\left(x^{2}-6x+9\right)
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.
y-1=ax^{2}-6ax+9a
Use the distributive property to multiply a by x^{2}-6x+9.
ax^{2}-6ax+9a=y-1
Swap sides so that all variable terms are on the left hand side.
\left(x^{2}-6x+9\right)a=y-1
Combine all terms containing a.
\frac{\left(x^{2}-6x+9\right)a}{x^{2}-6x+9}=\frac{y-1}{x^{2}-6x+9}
Divide both sides by x^{2}-6x+9.
a=\frac{y-1}{x^{2}-6x+9}
Dividing by x^{2}-6x+9 undoes the multiplication by x^{2}-6x+9.
a=\frac{y-1}{\left(x-3\right)^{2}}
Divide y-1 by x^{2}-6x+9.
Examples
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{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}