Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation x²-4x+3=0
-
Equation 12-y+2=0
-
Equation -2*x^2/(x^2-1)^2+1/(x^2-1)=0
-
Equation x^3+9*x^2+27*x+27=0
- Express {x} in terms of y where:
- -7*x-5*y=-17
- -4*x+19*y=-3
- -1*x-11*y=19
- -1*x-5*y=-4
-
Identical expressions
- (y+ eleven)^ two +(y- five)^ two =(five -y)(y+ eleven)
- (y plus 11) squared plus (y minus 5) squared equally (5 minus y)(y plus 11)
- (y plus eleven) to the power of two plus (y minus five) to the power of two equally (five minus y)(y plus eleven)
- (y+11)2+(y-5)2=(5-y)(y+11)
- y+112+y-52=5-yy+11
- (y+11)²+(y-5)²=(5-y)(y+11)
- (y+11) to the power of 2+(y-5) to the power of 2=(5-y)(y+11)
- y+11^2+y-5^2=5-yy+11
-
Similar expressions
- (y+11)^2+(y+5)^2=(5-y)(y+11)
- (y+11)^2+(y-5)^2=(5+y)(y+11)
- (y+11)^2-(y-5)^2=(5-y)(y+11)
- (y+11)^2+(y-5)^2=(5-y)(y-11)
- (y-11)^2+(y-5)^2=(5-y)(y+11)
(y+11)^2+(y-5)^2=(5-y)(y+11) equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
2 2 (y + 11) + (y - 5) = (5 - y)*(y + 11)
$$\left(y - 5\right)^{2} + \left(y + 11\right)^{2} = \left(5 - y\right) \left(y + 11\right)$$
Detail solution
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$\left(y - 5\right)^{2} + \left(y + 11\right)^{2} = \left(5 - y\right) \left(y + 11\right)$$
to
$$- \left(5 - y\right) \left(y + 11\right) + \left(\left(y - 5\right)^{2} + \left(y + 11\right)^{2}\right) = 0$$
Expand the expression in the equation
$$- \left(5 - y\right) \left(y + 11\right) + \left(\left(y - 5\right)^{2} + \left(y + 11\right)^{2}\right) = 0$$
We get the quadratic equation
$$3 y^{2} + 18 y + 91 = 0$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$y_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$y_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 3$$
$$b = 18$$
$$c = 91$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$y_{1} = -3 + \frac{8 \sqrt{3} i}{3}$$
$$y_{2} = -3 - \frac{8 \sqrt{3} i}{3}$$
left part with negative sign.
The equation is transformed from
$$\left(y - 5\right)^{2} + \left(y + 11\right)^{2} = \left(5 - y\right) \left(y + 11\right)$$
to
$$- \left(5 - y\right) \left(y + 11\right) + \left(\left(y - 5\right)^{2} + \left(y + 11\right)^{2}\right) = 0$$
Expand the expression in the equation
$$- \left(5 - y\right) \left(y + 11\right) + \left(\left(y - 5\right)^{2} + \left(y + 11\right)^{2}\right) = 0$$
We get the quadratic equation
$$3 y^{2} + 18 y + 91 = 0$$
This equation is of the form
a*y^2 + b*y + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$y_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$y_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 3$$
$$b = 18$$
$$c = 91$$
, then
D = b^2 - 4 * a * c =
(18)^2 - 4 * (3) * (91) = -768
Because D<0, then the equation
has no real roots,
but complex roots is exists.
y1 = (-b + sqrt(D)) / (2*a)
y2 = (-b - sqrt(D)) / (2*a)
or
$$y_{1} = -3 + \frac{8 \sqrt{3} i}{3}$$
$$y_{2} = -3 - \frac{8 \sqrt{3} i}{3}$$
Rapid solution
[src]
___
8*I*\/ 3
y1 = -3 - ---------
3
$$y_{1} = -3 - \frac{8 \sqrt{3} i}{3}$$
___
8*I*\/ 3
y2 = -3 + ---------
3
$$y_{2} = -3 + \frac{8 \sqrt{3} i}{3}$$
y2 = -3 + 8*sqrt(3)*i/3
Sum and product of roots
[src]
sum
___ ___
8*I*\/ 3 8*I*\/ 3
-3 - --------- + -3 + ---------
3 3
$$\left(-3 - \frac{8 \sqrt{3} i}{3}\right) + \left(-3 + \frac{8 \sqrt{3} i}{3}\right)$$
=
-6
$$-6$$
product
/ ___\ / ___\ | 8*I*\/ 3 | | 8*I*\/ 3 | |-3 - ---------|*|-3 + ---------| \ 3 / \ 3 /
$$\left(-3 - \frac{8 \sqrt{3} i}{3}\right) \left(-3 + \frac{8 \sqrt{3} i}{3}\right)$$
=
91/3
$$\frac{91}{3}$$
91/3
Numerical answer
[src]
y1 = -3.0 - 4.61880215351701*i
y2 = -3.0 + 4.61880215351701*i
y2 = -3.0 + 4.61880215351701*i