Solve left(2p+1right)x^2-left(7p+2right)x+7p-3=0

$$\left( 2p+1 \right) { x }^{ 2 } - \left( 7p+2 \right) x+7p-3 = 0$$

Solve for p (complex solution)

$$p=-\frac{\left(x-3\right)\left(x+1\right)}{2x^{2}-7x+7}$$

$$x\neq \frac{7+\sqrt{7}i}{4}\text{ and }x\neq \frac{-\sqrt{7}i+7}{4}$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2p+1 by x^{2}.

$$2px^{2}+x^{2}-\left(7p+2\right)x+7p-3=0$$

Use the distributive property to multiply 7p+2 by x.

$$2px^{2}+x^{2}-\left(7px+2x\right)+7p-3=0$$

To find the opposite of 7px+2x, find the opposite of each term.

$$2px^{2}+x^{2}-7px-2x+7p-3=0$$

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

$$2px^{2}-7px-2x+7p-3=-x^{2}$$

Add 2x to both sides.

$$2px^{2}-7px+7p-3=-x^{2}+2x$$

Add 3 to both sides.

$$2px^{2}-7px+7p=-x^{2}+2x+3$$

Combine all terms containing p.

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

The equation is in standard form.

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

Divide both sides by 2x^{2}-7x+7.

$$\frac{\left(2x^{2}-7x+7\right)p}{2x^{2}-7x+7}=-\frac{\left(x-3\right)\left(x+1\right)}{2x^{2}-7x+7}$$

Dividing by 2x^{2}-7x+7 undoes the multiplication by 2x^{2}-7x+7.

$$p=-\frac{\left(x-3\right)\left(x+1\right)}{2x^{2}-7x+7}$$

Solve for p

$$p=-\frac{\left(x-3\right)\left(x+1\right)}{2x^{2}-7x+7}$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2p+1 by x^{2}.

$$2px^{2}+x^{2}-\left(7p+2\right)x+7p-3=0$$

Use the distributive property to multiply 7p+2 by x.

$$2px^{2}+x^{2}-\left(7px+2x\right)+7p-3=0$$

To find the opposite of 7px+2x, find the opposite of each term.

$$2px^{2}+x^{2}-7px-2x+7p-3=0$$

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

$$2px^{2}-7px-2x+7p-3=-x^{2}$$

Add 2x to both sides.

$$2px^{2}-7px+7p-3=-x^{2}+2x$$

Add 3 to both sides.

$$2px^{2}-7px+7p=-x^{2}+2x+3$$

Combine all terms containing p.

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

The equation is in standard form.

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

Divide both sides by 2x^{2}-7x+7.

$$\frac{\left(2x^{2}-7x+7\right)p}{2x^{2}-7x+7}=-\frac{\left(x-3\right)\left(x+1\right)}{2x^{2}-7x+7}$$

Dividing by 2x^{2}-7x+7 undoes the multiplication by 2x^{2}-7x+7.

$$p=-\frac{\left(x-3\right)\left(x+1\right)}{2x^{2}-7x+7}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{\left(4-p\right)\left(7p+4\right)}+7p+2}{2\left(2p+1\right)}\text{; }x=\frac{-\sqrt{\left(4-p\right)\left(7p+4\right)}+7p+2}{2\left(2p+1\right)}\text{, }&p\neq -\frac{1}{2}\\x=\frac{13}{3}\text{, }&p=-\frac{1}{2}\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{\left(4-p\right)\left(7p+4\right)}+7p+2}{2\left(2p+1\right)}\text{; }x=\frac{-\sqrt{\left(4-p\right)\left(7p+4\right)}+7p+2}{2\left(2p+1\right)}\text{, }&p\neq -\frac{1}{2}\text{ and }p\geq -\frac{4}{7}\text{ and }p\leq 4\\x=\frac{13}{3}\text{, }&p=-\frac{1}{2}\end{matrix}\right.$$