Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation -15x=45
-
Equation x^2=15
-
Equation x^4=16
-
Equation 3/5*x=1
- Express {x} in terms of y where:
- 18*x+17*y=-14
- -1*x-6*y=18
- 16*x-17*y=-15
- -4*x-4*y=19
- Factor polynomial:
- x^2-15*x+63
-
Identical expressions
- x^ two - fifteen *x+ sixty-three = zero
- x squared minus 15 multiply by x plus 63 equally 0
- x to the power of two minus fifteen multiply by x plus sixty minus three equally zero
- x2-15*x+63=0
- x²-15*x+63=0
- x to the power of 2-15*x+63=0
- x^2-15x+63=0
- x2-15x+63=0
- x^2-15*x+63=O
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Similar expressions
- x^2+15*x+63=0
- x^2-15*x-63=0
x^2-15*x+63=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
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
$$a = 1$$
$$b = -15$$
$$c = 63$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = \frac{15}{2} + \frac{3 \sqrt{3} i}{2}$$
$$x_{2} = \frac{15}{2} - \frac{3 \sqrt{3} i}{2}$$
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
$$a = 1$$
$$b = -15$$
$$c = 63$$
, then
D = b^2 - 4 * a * c =
(-15)^2 - 4 * (1) * (63) = -27
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
$$x_{1} = \frac{15}{2} + \frac{3 \sqrt{3} i}{2}$$
$$x_{2} = \frac{15}{2} - \frac{3 \sqrt{3} i}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -15$$
$$q = \frac{c}{a}$$
$$q = 63$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 15$$
$$x_{1} x_{2} = 63$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -15$$
$$q = \frac{c}{a}$$
$$q = 63$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 15$$
$$x_{1} x_{2} = 63$$
Rapid solution
[src]
___
15 3*I*\/ 3
x1 = -- - ---------
2 2
$$x_{1} = \frac{15}{2} - \frac{3 \sqrt{3} i}{2}$$
___
15 3*I*\/ 3
x2 = -- + ---------
2 2
$$x_{2} = \frac{15}{2} + \frac{3 \sqrt{3} i}{2}$$
x2 = 15/2 + 3*sqrt(3)*i/2
Sum and product of roots
[src]
sum
___ ___ 15 3*I*\/ 3 15 3*I*\/ 3 -- - --------- + -- + --------- 2 2 2 2
$$\left(\frac{15}{2} - \frac{3 \sqrt{3} i}{2}\right) + \left(\frac{15}{2} + \frac{3 \sqrt{3} i}{2}\right)$$
=
15
$$15$$
product
/ ___\ / ___\ |15 3*I*\/ 3 | |15 3*I*\/ 3 | |-- - ---------|*|-- + ---------| \2 2 / \2 2 /
$$\left(\frac{15}{2} - \frac{3 \sqrt{3} i}{2}\right) \left(\frac{15}{2} + \frac{3 \sqrt{3} i}{2}\right)$$
=
63
$$63$$
63
Numerical answer
[src]
x1 = 7.5 - 2.59807621135332*i
x2 = 7.5 + 2.59807621135332*i
x2 = 7.5 + 2.59807621135332*i