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14x-17+3x^2=19+11x

14x-17+3x^2=19+11x equation

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Numerical solution:

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The solution

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14*x - 17 + 3*x  = 19 + 11*x
3x2+14x−17=11x+193 x^{2} + 14 x - 17 = 11 x + 19
Simplify
Detail solution
Move right part of the equation to
left part with negative sign.

The equation is transformed from
3x2+14x−17=11x+193 x^{2} + 14 x - 17 = 11 x + 19
to
(−11x−19)+(3x2+14x−17)=0\left(- 11 x - 19\right) + \left(3 x^{2} + 14 x - 17\right) = 0
This equation is of the form
a x2+b x+c=0a\ x^2 + b\ x + c = 0
A quadratic equation can be solved using the discriminant
The roots of the quadratic equation:
x1=D−b2ax_{1} = \frac{\sqrt{D} - b}{2 a}
x2=−D−b2ax_{2} = \frac{- \sqrt{D} - b}{2 a}
where D=b2−4acD = b^2 - 4 a c is the discriminant.
Because
a=3a = 3
b=3b = 3
c=−36c = -36
, then
D=b2−4 a c=D = b^2 - 4\ a\ c =
32−3⋅4(−36)=4413^{2} - 3 \cdot 4 \left(-36\right) = 441
Because D > 0, then the equation has two roots.
x1=(−b+D)2ax_1 = \frac{(-b + \sqrt{D})}{2 a}
x2=(−b−D)2ax_2 = \frac{(-b - \sqrt{D})}{2 a}
or
x1=3x_{1} = 3
Simplify
x2=−4x_{2} = -4
Simplify
Vieta's Theorem
rewrite the equation
3x2+14x−17=11x+193 x^{2} + 14 x - 17 = 11 x + 19
of
ax2+bx+c=0a x^{2} + b x + c = 0
as reduced quadratic equation
x2+bxa+ca=0x^{2} + \frac{b x}{a} + \frac{c}{a} = 0
x2+x−12=0x^{2} + x - 12 = 0
px+x2+q=0p x + x^{2} + q = 0
where
p=bap = \frac{b}{a}
p=1p = 1
q=caq = \frac{c}{a}
q=−12q = -12
Vieta Formulas
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=−1x_{1} + x_{2} = -1
x1x2=−12x_{1} x_{2} = -12
The graph
-12.5-10.0-7.5-5.0-2.50.02.55.07.510.012.5-200200
Plot the graph f1(x)
Plot the graph f2(x)
Sum and product of roots [src] 
sum
-4 + 3
(−4)+(3)\left(-4\right) + \left(3\right) 
=
-1
−1-1
Simplify 
product
-4 * 3
(−4)∗(3)\left(-4\right) * \left(3\right) 
=
-12
−12-12
Simplify
Rapid solution [src] 
x_1 = -4
x1=−4x_{1} = -4
Simplify 
x_2 = 3
x2=3x_{2} = 3
Simplify
Numerical answer [src] 
x1 = 3.0
x2 = -4.0
x2 = -4.0
The graph
14x-17+3x^2=19+11x equation
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