Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation 15*x^2-4*x-3=0
- Equation 64-x^2=0
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Equation 16x^2-1=0
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Equation -9*x=36
- Express {x} in terms of y where:
- -8*x-2*y=-11
- -20*x-12*y=-17
- 3*x+8*y=-7
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- Factor squared:
- x^2+x-5
-
Identical expressions
- x^ two +x- five = zero
- x squared plus x minus 5 equally 0
- x to the power of two plus x minus five equally zero
- x2+x-5=0
- x²+x-5=0
- x to the power of 2+x-5=0
- x^2+x-5=O
-
Similar expressions
- x^2-x-5=0
- x^2+x+5=0
x^2+x-5=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
$$a = 1$$
$$b = 1$$
$$c = -5$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = - \frac{1}{2} + \frac{\sqrt{21}}{2}$$
$$x_{2} = - \frac{\sqrt{21}}{2} - \frac{1}{2}$$
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
$$a = 1$$
$$b = 1$$
$$c = -5$$
, then
D = b^2 - 4 * a * c =
(1)^2 - 4 * (1) * (-5) = 21
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = - \frac{1}{2} + \frac{\sqrt{21}}{2}$$
$$x_{2} = - \frac{\sqrt{21}}{2} - \frac{1}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 1$$
$$q = \frac{c}{a}$$
$$q = -5$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -1$$
$$x_{1} x_{2} = -5$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 1$$
$$q = \frac{c}{a}$$
$$q = -5$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -1$$
$$x_{1} x_{2} = -5$$
Sum and product of roots
[src]
sum
____ ____ 1 \/ 21 1 \/ 21 - - + ------ + - - - ------ 2 2 2 2
$$\left(- \frac{\sqrt{21}}{2} - \frac{1}{2}\right) + \left(- \frac{1}{2} + \frac{\sqrt{21}}{2}\right)$$
=
-1
$$-1$$
product
/ ____\ / ____\ | 1 \/ 21 | | 1 \/ 21 | |- - + ------|*|- - - ------| \ 2 2 / \ 2 2 /
$$\left(- \frac{1}{2} + \frac{\sqrt{21}}{2}\right) \left(- \frac{\sqrt{21}}{2} - \frac{1}{2}\right)$$
=
-5
$$-5$$
-5
Rapid solution
[src]
____
1 \/ 21
x1 = - - + ------
2 2
$$x_{1} = - \frac{1}{2} + \frac{\sqrt{21}}{2}$$
____
1 \/ 21
x2 = - - - ------
2 2
$$x_{2} = - \frac{\sqrt{21}}{2} - \frac{1}{2}$$
x2 = -sqrt(21)/2 - 1/2
The graph