Mister Exam

Other calculators

4x²-5x-7=0 equation

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

v

Numerical solution:

Do search numerical solution at [, ]

The solution

You have entered [src] 
   2              
4*x  - 5*x - 7 = 0
(4x2−5x)−7=0\left(4 x^{2} - 5 x\right) - 7 = 0
Simplify
Detail solution
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:
x1=D−b2ax_{1} = \frac{\sqrt{D} - b}{2 a}
x2=−D−b2ax_{2} = \frac{- \sqrt{D} - b}{2 a}
where D = b^2 - 4*a*c - it is the discriminant.
Because
a=4a = 4
b=−5b = -5
c=−7c = -7
, then
D = b^2 - 4 * a * c = 

(-5)^2 - 4 * (4) * (-7) = 137

Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)

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

or
x1=58+1378x_{1} = \frac{5}{8} + \frac{\sqrt{137}}{8}
x2=58−1378x_{2} = \frac{5}{8} - \frac{\sqrt{137}}{8}
Vieta's Theorem
rewrite the equation
(4x2−5x)−7=0\left(4 x^{2} - 5 x\right) - 7 = 0
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
x2−5x4−74=0x^{2} - \frac{5 x}{4} - \frac{7}{4} = 0
px+q+x2=0p x + q + x^{2} = 0
where
p=bap = \frac{b}{a}
p=−54p = - \frac{5}{4}
q=caq = \frac{c}{a}
q=−74q = - \frac{7}{4}
Vieta Formulas
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=54x_{1} + x_{2} = \frac{5}{4}
x1x2=−74x_{1} x_{2} = - \frac{7}{4}
The graph
05-15-10-51015-5001000
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
           _____
     5   \/ 137 
x1 = - - -------
     8      8
x1=58−1378x_{1} = \frac{5}{8} - \frac{\sqrt{137}}{8} 
           _____
     5   \/ 137 
x2 = - + -------
     8      8
x2=58+1378x_{2} = \frac{5}{8} + \frac{\sqrt{137}}{8} 
x2 = 5/8 + sqrt(137)/8
Sum and product of roots [src] 
sum
      _____         _____
5   \/ 137    5   \/ 137 
- - ------- + - + -------
8      8      8      8
(58−1378)+(58+1378)\left(\frac{5}{8} - \frac{\sqrt{137}}{8}\right) + \left(\frac{5}{8} + \frac{\sqrt{137}}{8}\right) 
=
5/4
54\frac{5}{4} 
product
/      _____\ /      _____\
|5   \/ 137 | |5   \/ 137 |
|- - -------|*|- + -------|
\8      8   / \8      8   /
(58−1378)(58+1378)\left(\frac{5}{8} - \frac{\sqrt{137}}{8}\right) \left(\frac{5}{8} + \frac{\sqrt{137}}{8}\right) 
=
-7/4
−74- \frac{7}{4} 
-7/4
Numerical answer [src] 
x1 = -0.838087488839953
x2 = 2.08808748883995
x2 = 2.08808748883995
    © Mister Exam – Calculator