Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation 49*x^3+14*x^2+x=0
-
Equation 4^x=8
-
Equation x^2-x=12
-
Equation x-5(x+3)=5
- Express {x} in terms of y where:
- -7*x-13*y=-12
- -3*x+19*y=16
- -4*x+4*y=-9
- -19*x-2*y=-6
-
Identical expressions
- two x^2- fourteen *x+ ten = eight
- 2x squared minus 14 multiply by x plus 10 equally 8
- two x squared minus fourteen multiply by x plus ten equally eight
- 2x2-14*x+10=8
- 2x²-14*x+10=8
- 2x to the power of 2-14*x+10=8
- 2x^2-14x+10=8
- 2x2-14x+10=8
-
Similar expressions
- 2x^2+14*x+10=8
- 2x^2-14*x-10=8
2x^2-14*x+10=8 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
$$\left(2 x^{2} - 14 x\right) + 10 = 8$$
to
$$\left(\left(2 x^{2} - 14 x\right) + 10\right) - 8 = 0$$
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
$$a = 2$$
$$b = -14$$
$$c = 2$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{3 \sqrt{5}}{2} + \frac{7}{2}$$
$$x_{2} = \frac{7}{2} - \frac{3 \sqrt{5}}{2}$$
left part with negative sign.
The equation is transformed from
$$\left(2 x^{2} - 14 x\right) + 10 = 8$$
to
$$\left(\left(2 x^{2} - 14 x\right) + 10\right) - 8 = 0$$
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
$$a = 2$$
$$b = -14$$
$$c = 2$$
, then
D = b^2 - 4 * a * c =
(-14)^2 - 4 * (2) * (2) = 180
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{3 \sqrt{5}}{2} + \frac{7}{2}$$
$$x_{2} = \frac{7}{2} - \frac{3 \sqrt{5}}{2}$$
Vieta's Theorem
rewrite the equation
$$\left(2 x^{2} - 14 x\right) + 10 = 8$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - 7 x + 1 = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -7$$
$$q = \frac{c}{a}$$
$$q = 1$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 7$$
$$x_{1} x_{2} = 1$$
$$\left(2 x^{2} - 14 x\right) + 10 = 8$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - 7 x + 1 = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -7$$
$$q = \frac{c}{a}$$
$$q = 1$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 7$$
$$x_{1} x_{2} = 1$$
Sum and product of roots
[src]
sum
___ ___ 7 3*\/ 5 7 3*\/ 5 - - ------- + - + ------- 2 2 2 2
$$\left(\frac{7}{2} - \frac{3 \sqrt{5}}{2}\right) + \left(\frac{3 \sqrt{5}}{2} + \frac{7}{2}\right)$$
=
7
$$7$$
product
/ ___\ / ___\ |7 3*\/ 5 | |7 3*\/ 5 | |- - -------|*|- + -------| \2 2 / \2 2 /
$$\left(\frac{7}{2} - \frac{3 \sqrt{5}}{2}\right) \left(\frac{3 \sqrt{5}}{2} + \frac{7}{2}\right)$$
=
1
$$1$$
1
Rapid solution
[src]
___
7 3*\/ 5
x1 = - - -------
2 2
$$x_{1} = \frac{7}{2} - \frac{3 \sqrt{5}}{2}$$
___
7 3*\/ 5
x2 = - + -------
2 2
$$x_{2} = \frac{3 \sqrt{5}}{2} + \frac{7}{2}$$
x2 = 3*sqrt(5)/2 + 7/2