Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation z^2+z+2=0
-
Equation 4*x^2-13*x+3=0
-
Equation 3*x^2+6*x+3=0
-
Equation 3x+5=0
- Express {x} in terms of y where:
- 2*x+11*y=15
- 2*x+14*y=-13
- 8*x+5*y=-1
- -7*x+3*y=2
-
Identical expressions
- x= forty-two +(four /7xx)
- x equally 42 plus (4 divide by 7xx)
- x equally forty minus two plus (four divide by 7xx)
- x=42+4/7xx
- x=42+(4 divide by 7xx)
-
Similar expressions
- x=42-(4/7xx)
x=42+(4/7xx) equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$x = \frac{4 x}{7} x + 42$$
to
$$x + \left(- \frac{4 x}{7} x - 42\right) = 0$$
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
$$a = - \frac{4}{7}$$
$$b = 1$$
$$c = -42$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = \frac{7}{8} - \frac{7 \sqrt{95} i}{8}$$
$$x_{2} = \frac{7}{8} + \frac{7 \sqrt{95} i}{8}$$
left part with negative sign.
The equation is transformed from
$$x = \frac{4 x}{7} x + 42$$
to
$$x + \left(- \frac{4 x}{7} x - 42\right) = 0$$
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
$$a = - \frac{4}{7}$$
$$b = 1$$
$$c = -42$$
, then
D = b^2 - 4 * a * c =
(1)^2 - 4 * (-4/7) * (-42) = -95
Because D<0, then the equation
has no real roots,
but complex roots is exists.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{7}{8} - \frac{7 \sqrt{95} i}{8}$$
$$x_{2} = \frac{7}{8} + \frac{7 \sqrt{95} i}{8}$$
Vieta's Theorem
rewrite the equation
$$x = \frac{4 x}{7} x + 42$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{7 x}{4} + \frac{147}{2} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{7}{4}$$
$$q = \frac{c}{a}$$
$$q = \frac{147}{2}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{7}{4}$$
$$x_{1} x_{2} = \frac{147}{2}$$
$$x = \frac{4 x}{7} x + 42$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{7 x}{4} + \frac{147}{2} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{7}{4}$$
$$q = \frac{c}{a}$$
$$q = \frac{147}{2}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{7}{4}$$
$$x_{1} x_{2} = \frac{147}{2}$$
Sum and product of roots
[src]
sum
____ ____ 7 7*I*\/ 95 7 7*I*\/ 95 - - ---------- + - + ---------- 8 8 8 8
$$\left(\frac{7}{8} - \frac{7 \sqrt{95} i}{8}\right) + \left(\frac{7}{8} + \frac{7 \sqrt{95} i}{8}\right)$$
=
7/4
$$\frac{7}{4}$$
product
/ ____\ / ____\ |7 7*I*\/ 95 | |7 7*I*\/ 95 | |- - ----------|*|- + ----------| \8 8 / \8 8 /
$$\left(\frac{7}{8} - \frac{7 \sqrt{95} i}{8}\right) \left(\frac{7}{8} + \frac{7 \sqrt{95} i}{8}\right)$$
=
147/2
$$\frac{147}{2}$$
147/2
Rapid solution
[src]
____
7 7*I*\/ 95
x1 = - - ----------
8 8
$$x_{1} = \frac{7}{8} - \frac{7 \sqrt{95} i}{8}$$
____
7 7*I*\/ 95
x2 = - + ----------
8 8
$$x_{2} = \frac{7}{8} + \frac{7 \sqrt{95} i}{8}$$
x2 = 7/8 + 7*sqrt(95)*i/8
Numerical answer
[src]
x1 = 0.875 + 8.52844505170784*i
x2 = 0.875 - 8.52844505170784*i
x2 = 0.875 - 8.52844505170784*i