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6x^2-42x+72=6(x-4) equation
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The solution
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2 6*x - 42*x + 72 = 6*(x - 4)
$$\left(6 x^{2} - 42 x\right) + 72 = 6 \left(x - 4\right)$$
Detail solution
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$\left(6 x^{2} - 42 x\right) + 72 = 6 \left(x - 4\right)$$
to
$$- 6 \left(x - 4\right) + \left(\left(6 x^{2} - 42 x\right) + 72\right) = 0$$
Expand the expression in the equation
$$- 6 \left(x - 4\right) + \left(\left(6 x^{2} - 42 x\right) + 72\right) = 0$$
We get the quadratic equation
$$6 x^{2} - 48 x + 96 = 0$$
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
$$a = 6$$
$$b = -48$$
$$c = 96$$
, then
Because D = 0, then the equation has one root.
$$x_{1} = 4$$
left part with negative sign.
The equation is transformed from
$$\left(6 x^{2} - 42 x\right) + 72 = 6 \left(x - 4\right)$$
to
$$- 6 \left(x - 4\right) + \left(\left(6 x^{2} - 42 x\right) + 72\right) = 0$$
Expand the expression in the equation
$$- 6 \left(x - 4\right) + \left(\left(6 x^{2} - 42 x\right) + 72\right) = 0$$
We get the quadratic equation
$$6 x^{2} - 48 x + 96 = 0$$
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
$$a = 6$$
$$b = -48$$
$$c = 96$$
, then
D = b^2 - 4 * a * c =
(-48)^2 - 4 * (6) * (96) = 0
Because D = 0, then the equation has one root.
x = -b/2a = --48/2/(6)
$$x_{1} = 4$$
Vieta's Theorem
rewrite the equation
$$\left(6 x^{2} - 42 x\right) + 72 = 6 \left(x - 4\right)$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - 8 x + 16 = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -8$$
$$q = \frac{c}{a}$$
$$q = 16$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 8$$
$$x_{1} x_{2} = 16$$
$$\left(6 x^{2} - 42 x\right) + 72 = 6 \left(x - 4\right)$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - 8 x + 16 = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -8$$
$$q = \frac{c}{a}$$
$$q = 16$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 8$$
$$x_{1} x_{2} = 16$$