The teacher will be very surprised to see your correct solution 😉

Detail solution

Move right part of the equation to

left part with negative sign.

The equation is transformed from

$$3 x^{2} + 14 x - 17 = 11 x + 19$$

to

$$\left(- 11 x - 19\right) + \left(3 x^{2} + 14 x - 17\right) = 0$$

This equation is of the form

$$a\ x^2 + b\ x + c = 0$$

A quadratic equation can be solved using the discriminant

The roots of the quadratic equation:

$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$

$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$

where $D = b^2 - 4 a c$ is the discriminant.

Because

$$a = 3$$

$$b = 3$$

$$c = -36$$

, then

$$D = b^2 - 4\ a\ c = $$

$$3^{2} - 3 \cdot 4 \left(-36\right) = 441$$

Because D > 0, then the equation has two roots.

$$x_1 = \frac{(-b + \sqrt{D})}{2 a}$$

$$x_2 = \frac{(-b - \sqrt{D})}{2 a}$$

or

$$x_{1} = 3$$

Simplify

$$x_{2} = -4$$

Simplify

left part with negative sign.

The equation is transformed from

$$3 x^{2} + 14 x - 17 = 11 x + 19$$

to

$$\left(- 11 x - 19\right) + \left(3 x^{2} + 14 x - 17\right) = 0$$

This equation is of the form

$$a\ x^2 + b\ x + c = 0$$

A quadratic equation can be solved using the discriminant

The roots of the quadratic equation:

$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$

$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$

where $D = b^2 - 4 a c$ is the discriminant.

Because

$$a = 3$$

$$b = 3$$

$$c = -36$$

, then

$$D = b^2 - 4\ a\ c = $$

$$3^{2} - 3 \cdot 4 \left(-36\right) = 441$$

Because D > 0, then the equation has two roots.

$$x_1 = \frac{(-b + \sqrt{D})}{2 a}$$

$$x_2 = \frac{(-b - \sqrt{D})}{2 a}$$

or

$$x_{1} = 3$$

Simplify

$$x_{2} = -4$$

Simplify

Vieta's Theorem

rewrite the equation

$$3 x^{2} + 14 x - 17 = 11 x + 19$$

of

$$a x^{2} + b x + c = 0$$

as reduced quadratic equation

$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$

$$x^{2} + x - 12 = 0$$

$$p x + x^{2} + q = 0$$

where

$$p = \frac{b}{a}$$

$$p = 1$$

$$q = \frac{c}{a}$$

$$q = -12$$

Vieta Formulas

$$x_{1} + x_{2} = - p$$

$$x_{1} x_{2} = q$$

$$x_{1} + x_{2} = -1$$

$$x_{1} x_{2} = -12$$

$$3 x^{2} + 14 x - 17 = 11 x + 19$$

of

$$a x^{2} + b x + c = 0$$

as reduced quadratic equation

$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$

$$x^{2} + x - 12 = 0$$

$$p x + x^{2} + q = 0$$

where

$$p = \frac{b}{a}$$

$$p = 1$$

$$q = \frac{c}{a}$$

$$q = -12$$

Vieta Formulas

$$x_{1} + x_{2} = - p$$

$$x_{1} x_{2} = q$$

$$x_{1} + x_{2} = -1$$

$$x_{1} x_{2} = -12$$

Sum and product of roots
[src]

sum

-4 + 3

$$\left(-4\right) + \left(3\right)$$

=

-1

$$-1$$

product

-4 * 3

$$\left(-4\right) * \left(3\right)$$

=

-12

$$-12$$

The graph