Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation x^2-x=12
-
Equation x-5(x+3)=5
-
Equation (|x+4|)=5
-
Equation -x=2
- Express {x} in terms of y where:
- -4*x+4*y=-9
- -19*x-2*y=-6
- -2*x+2*y=4
- -17*x-8*y=7
-
Identical expressions
- 13²-x²=14²-(fifteen -x²)
- 13² minus x² equally 14² minus (15 minus x²)
- 13² minus x² equally 14² minus (fifteen minus x²)
- 13²-x²=14²-15-x²
-
Similar expressions
- 13²+x²=14²-(15-x²)
- 13²-x²=14²-(15+x²)
- 13²-x²=14²+(15-x²)
13²-x²=14²-(15-x²) 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
$$169 - x^{2} = \left(x^{2} - 15\right) + 196$$
to
$$\left(169 - x^{2}\right) + \left(\left(15 - x^{2}\right) - 196\right) = 0$$
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
$$a = -2$$
$$b = 0$$
$$c = -12$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = - \sqrt{6} i$$
$$x_{2} = \sqrt{6} i$$
left part with negative sign.
The equation is transformed from
$$169 - x^{2} = \left(x^{2} - 15\right) + 196$$
to
$$\left(169 - x^{2}\right) + \left(\left(15 - x^{2}\right) - 196\right) = 0$$
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
$$a = -2$$
$$b = 0$$
$$c = -12$$
, then
D = b^2 - 4 * a * c =
(0)^2 - 4 * (-2) * (-12) = -96
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
$$x_{1} = - \sqrt{6} i$$
$$x_{2} = \sqrt{6} i$$
Vieta's Theorem
rewrite the equation
$$169 - x^{2} = \left(x^{2} - 15\right) + 196$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + 6 = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 6$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 0$$
$$x_{1} x_{2} = 6$$
$$169 - x^{2} = \left(x^{2} - 15\right) + 196$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + 6 = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 6$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 0$$
$$x_{1} x_{2} = 6$$
Rapid solution
[src]
___ x1 = -I*\/ 6
$$x_{1} = - \sqrt{6} i$$
___ x2 = I*\/ 6
$$x_{2} = \sqrt{6} i$$
x2 = sqrt(6)*i
Sum and product of roots
[src]
sum
___ ___ - I*\/ 6 + I*\/ 6
$$- \sqrt{6} i + \sqrt{6} i$$
=
0
$$0$$
product
___ ___ -I*\/ 6 *I*\/ 6
$$- \sqrt{6} i \sqrt{6} i$$
=
6
$$6$$
6