Mister Exam
Other calculators
-
How to use it?
- Equation:
- Equation x^3-64*x=0
-
Equation 3x^2=2x+x^3
-
Equation (x^2+3)/(x^2+1)=2
- Equation x^3+2*x^2-36*x-72=0
- Express {x} in terms of y where:
- -2*x-5*y=16
- -5*x-14*y=6
- 6*x+10*y=-19
- 12*x+20*y=20
- Graphing y =:
- 4x^2-2x-3
-
Identical expressions
- 4x^ two -2x- three
- 4x squared minus 2x minus 3
- 4x to the power of two minus 2x minus three
- 4x2-2x-3
- 4x²-2x-3
- 4x to the power of 2-2x-3
-
Similar expressions
- 4x^2-2x+3
- 4x^2+2x-3
4x^2-2x-3 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
$$a = 4$$
$$b = -2$$
$$c = -3$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{1}{4} + \frac{\sqrt{13}}{4}$$
$$x_{2} = \frac{1}{4} - \frac{\sqrt{13}}{4}$$
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
$$a = 4$$
$$b = -2$$
$$c = -3$$
, then
D = b^2 - 4 * a * c =
(-2)^2 - 4 * (4) * (-3) = 52
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{1}{4} + \frac{\sqrt{13}}{4}$$
$$x_{2} = \frac{1}{4} - \frac{\sqrt{13}}{4}$$
Vieta's Theorem
rewrite the equation
$$\left(4 x^{2} - 2 x\right) - 3 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{x}{2} - \frac{3}{4} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{1}{2}$$
$$q = \frac{c}{a}$$
$$q = - \frac{3}{4}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{1}{2}$$
$$x_{1} x_{2} = - \frac{3}{4}$$
$$\left(4 x^{2} - 2 x\right) - 3 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{x}{2} - \frac{3}{4} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{1}{2}$$
$$q = \frac{c}{a}$$
$$q = - \frac{3}{4}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{1}{2}$$
$$x_{1} x_{2} = - \frac{3}{4}$$
Rapid solution
[src]
____
1 \/ 13
x1 = - - ------
4 4
$$x_{1} = \frac{1}{4} - \frac{\sqrt{13}}{4}$$
____
1 \/ 13
x2 = - + ------
4 4
$$x_{2} = \frac{1}{4} + \frac{\sqrt{13}}{4}$$
x2 = 1/4 + sqrt(13)/4
Sum and product of roots
[src]
sum
____ ____ 1 \/ 13 1 \/ 13 - - ------ + - + ------ 4 4 4 4
$$\left(\frac{1}{4} - \frac{\sqrt{13}}{4}\right) + \left(\frac{1}{4} + \frac{\sqrt{13}}{4}\right)$$
=
1/2
$$\frac{1}{2}$$
product
/ ____\ / ____\ |1 \/ 13 | |1 \/ 13 | |- - ------|*|- + ------| \4 4 / \4 4 /
$$\left(\frac{1}{4} - \frac{\sqrt{13}}{4}\right) \left(\frac{1}{4} + \frac{\sqrt{13}}{4}\right)$$
=
-3/4
$$- \frac{3}{4}$$
-3/4