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Detail solution

Expand the expression in the equation

$$\left(x - 5\right) \left(x - 1\right) - 21 = 0$$

We get the quadratic equation

$$x^{2} - 6 x - 16 = 0$$

This equation is of the form

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 - it is the discriminant.

Because

$$a = 1$$

$$b = -6$$

$$c = -16$$

, then

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

or

$$x_{1} = 8$$

$$x_{2} = -2$$

$$\left(x - 5\right) \left(x - 1\right) - 21 = 0$$

We get the quadratic equation

$$x^{2} - 6 x - 16 = 0$$

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 - it is the discriminant.

Because

$$a = 1$$

$$b = -6$$

$$c = -16$$

, then

D = b^2 - 4 * a * c =

(-6)^2 - 4 * (1) * (-16) = 100

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

x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

or

$$x_{1} = 8$$

$$x_{2} = -2$$

Sum and product of roots
[src]

sum

-2 + 8

$$-2 + 8$$

=

6

$$6$$

product

-2*8

$$- 16$$

=

-16

$$-16$$

-16

The graph