Solve ax^2+x^2-4x+3a-7=0

$$a x ^ { 2 } + x ^ { 2 } - 4 x + 3 a - 7 = 0$$

Solve for a (complex solution)

$$a=-\frac{x^{2}-4x-7}{x^{2}+3}$$

$$x\neq -\sqrt{3}i\text{ and }x\neq \sqrt{3}i$$

Steps for Solving Linear Equation

Subtract x^{2} from both sides. Anything subtracted from zero gives its negation.

$$ax^{2}-4x+3a-7=-x^{2}$$

Add 4x to both sides.

$$ax^{2}+3a-7=-x^{2}+4x$$

Add 7 to both sides.

$$ax^{2}+3a=-x^{2}+4x+7$$

Combine all terms containing a.

$$\left(x^{2}+3\right)a=-x^{2}+4x+7$$

The equation is in standard form.

$$\left(x^{2}+3\right)a=7+4x-x^{2}$$

Divide both sides by x^{2}+3.

$$\frac{\left(x^{2}+3\right)a}{x^{2}+3}=\frac{7+4x-x^{2}}{x^{2}+3}$$

Dividing by x^{2}+3 undoes the multiplication by x^{2}+3.

$$a=\frac{7+4x-x^{2}}{x^{2}+3}$$

Solve for a

$$a=-\frac{x^{2}-4x-7}{x^{2}+3}$$

Steps for Solving Linear Equation

Subtract x^{2} from both sides. Anything subtracted from zero gives its negation.

$$ax^{2}-4x+3a-7=-x^{2}$$

Add 4x to both sides.

$$ax^{2}+3a-7=-x^{2}+4x$$

Add 7 to both sides.

$$ax^{2}+3a=-x^{2}+4x+7$$

Combine all terms containing a.

$$\left(x^{2}+3\right)a=-x^{2}+4x+7$$

The equation is in standard form.

$$\left(x^{2}+3\right)a=7+4x-x^{2}$$

Divide both sides by x^{2}+3.

$$\frac{\left(x^{2}+3\right)a}{x^{2}+3}=\frac{7+4x-x^{2}}{x^{2}+3}$$

Dividing by x^{2}+3 undoes the multiplication by x^{2}+3.

$$a=\frac{7+4x-x^{2}}{x^{2}+3}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{11+4a-3a^{2}}+2}{a+1}\text{; }x=\frac{-\sqrt{11+4a-3a^{2}}+2}{a+1}\text{, }&a\neq -1\\x=-\frac{5}{2}\text{, }&a=-1\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{11+4a-3a^{2}}+2}{a+1}\text{; }x=\frac{-\sqrt{11+4a-3a^{2}}+2}{a+1}\text{, }&a\neq -1\text{ and }a\geq \frac{2-\sqrt{37}}{3}\text{ and }a\leq \frac{\sqrt{37}+2}{3}\\x=-\frac{5}{2}\text{, }&a=-1\end{matrix}\right.$$