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x^2-5x-12=0

x^2-5x-12=0 equation

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Numerical solution:

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The solution

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 2               
x  - 5*x - 12 = 0
x2−5x−12=0x^{2} - 5 x - 12 = 0
Simplify
Detail solution
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=1a = 1
b=−5b = -5
c=−12c = -12
, then
D=b2−4 a c=D = b^2 - 4\ a\ c =
(−5)2−1⋅4(−12)=73\left(-5\right)^{2} - 1 \cdot 4 \left(-12\right) = 73
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=52+732x_{1} = \frac{5}{2} + \frac{\sqrt{73}}{2}
Simplify
x2=−732+52x_{2} = - \frac{\sqrt{73}}{2} + \frac{5}{2}
Simplify
Vieta's Theorem
it is reduced quadratic equation
px+x2+q=0p x + x^{2} + q = 0
where
p=bap = \frac{b}{a}
p=−5p = -5
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=5x_{1} + x_{2} = 5
x1x2=−12x_{1} x_{2} = -12
The graph
05-15-10-510152025-100100
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
            ____
      5   \/ 73 
x_1 = - - ------
      2     2
x1=−732+52x_{1} = - \frac{\sqrt{73}}{2} + \frac{5}{2}
Simplify 
            ____
      5   \/ 73 
x_2 = - + ------
      2     2
x2=52+732x_{2} = \frac{5}{2} + \frac{\sqrt{73}}{2}
Simplify
Sum and product of roots [src] 
sum
      ____         ____
5   \/ 73    5   \/ 73 
- - ------ + - + ------
2     2      2     2
(−732+52)+(52+732)\left(- \frac{\sqrt{73}}{2} + \frac{5}{2}\right) + \left(\frac{5}{2} + \frac{\sqrt{73}}{2}\right) 
=
5
55
Simplify 
product
      ____         ____
5   \/ 73    5   \/ 73 
- - ------ * - + ------
2     2      2     2
(−732+52)∗(52+732)\left(- \frac{\sqrt{73}}{2} + \frac{5}{2}\right) * \left(\frac{5}{2} + \frac{\sqrt{73}}{2}\right) 
=
-12
−12-12
Simplify
Numerical answer [src] 
x1 = 6.77200187265877
x2 = -1.77200187265877
x2 = -1.77200187265877
The graph
x^2-5x-12=0 equation
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