Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation 2x-3(x+3)=-5
-
Equation 2x-3=4
-
Equation 6=8-5(7x-1)
-
Equation 5(x-2)^2=-6x-44
- Express {x} in terms of y where:
- -14*x+10*y=19
- 5*x-14*y=-15
- -7*x+15*y=-11
- -18*x-14*y=12
-
Identical expressions
- x^ two -5x- seven = zero
- x squared minus 5x minus 7 equally 0
- x to the power of two minus 5x minus seven equally zero
- x2-5x-7=0
- x²-5x-7=0
- x to the power of 2-5x-7=0
- x^2-5x-7=O
-
Similar expressions
- x^2+5x-7=0
- x^2-5x+7=0
x^2-5x-7=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
$$a = 1$$
$$b = -5$$
$$c = -7$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{5}{2} + \frac{\sqrt{53}}{2}$$
$$x_{2} = \frac{5}{2} - \frac{\sqrt{53}}{2}$$
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
$$a = 1$$
$$b = -5$$
$$c = -7$$
, then
D = b^2 - 4 * a * c =
(-5)^2 - 4 * (1) * (-7) = 53
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{5}{2} + \frac{\sqrt{53}}{2}$$
$$x_{2} = \frac{5}{2} - \frac{\sqrt{53}}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -5$$
$$q = \frac{c}{a}$$
$$q = -7$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 5$$
$$x_{1} x_{2} = -7$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -5$$
$$q = \frac{c}{a}$$
$$q = -7$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 5$$
$$x_{1} x_{2} = -7$$
Rapid solution
[src]
____
5 \/ 53
x1 = - - ------
2 2
$$x_{1} = \frac{5}{2} - \frac{\sqrt{53}}{2}$$
____
5 \/ 53
x2 = - + ------
2 2
$$x_{2} = \frac{5}{2} + \frac{\sqrt{53}}{2}$$
x2 = 5/2 + sqrt(53)/2
Sum and product of roots
[src]
sum
____ ____ 5 \/ 53 5 \/ 53 - - ------ + - + ------ 2 2 2 2
$$\left(\frac{5}{2} - \frac{\sqrt{53}}{2}\right) + \left(\frac{5}{2} + \frac{\sqrt{53}}{2}\right)$$
=
5
$$5$$
product
/ ____\ / ____\ |5 \/ 53 | |5 \/ 53 | |- - ------|*|- + ------| \2 2 / \2 2 /
$$\left(\frac{5}{2} - \frac{\sqrt{53}}{2}\right) \left(\frac{5}{2} + \frac{\sqrt{53}}{2}\right)$$
=
-7
$$-7$$
-7