Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation x^4-2x^2-8=0
-
Equation 5x-25+2x²=17+13x
- Equation sin(pi(x+4)/6)=-0.5
-
Equation 8*cos(x)+sin(7*x)-16*x=x^3+8
- Express {x} in terms of y where:
- 4*x-16*y=6
- -14*x-1*y=3
- 8*x+7*y=-13
- 5*x+5*y=20
-
Identical expressions
- 5x- twenty-five +2x²= seventeen +13x
- 5x minus 25 plus 2x² equally 17 plus 13x
- 5x minus twenty minus five plus 2x² equally seventeen plus 13x
-
Similar expressions
- 5x+25+2x²=17+13x
- 5x-25+2x²=17-13x
- 5x-25-2x²=17+13x
5x-25+2x²=17+13x 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
$$2 x^{2} + 5 x - 25 = 13 x + 17$$
to
$$\left(- 13 x - 17\right) + \left(2 x^{2} + 5 x - 25\right) = 0$$
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 $D = b^2 - 4 a c$ is the discriminant.
Because
$$a = 2$$
$$b = -8$$
$$c = -42$$
, then
$$D = b^2 - 4\ a\ c = $$
$$\left(-8\right)^{2} - 2 \cdot 4 \left(-42\right) = 400$$
Because D > 0, then the equation has two roots.
$$x_1 = \frac{(-b + \sqrt{D})}{2 a}$$
$$x_2 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$x_{1} = 7$$
Simplify
$$x_{2} = -3$$
Simplify
left part with negative sign.
The equation is transformed from
$$2 x^{2} + 5 x - 25 = 13 x + 17$$
to
$$\left(- 13 x - 17\right) + \left(2 x^{2} + 5 x - 25\right) = 0$$
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 $D = b^2 - 4 a c$ is the discriminant.
Because
$$a = 2$$
$$b = -8$$
$$c = -42$$
, then
$$D = b^2 - 4\ a\ c = $$
$$\left(-8\right)^{2} - 2 \cdot 4 \left(-42\right) = 400$$
Because D > 0, then the equation has two roots.
$$x_1 = \frac{(-b + \sqrt{D})}{2 a}$$
$$x_2 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$x_{1} = 7$$
Simplify
$$x_{2} = -3$$
Simplify
Vieta's Theorem
rewrite the equation
$$2 x^{2} + 5 x - 25 = 13 x + 17$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - 4 x - 21 = 0$$
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = -4$$
$$q = \frac{c}{a}$$
$$q = -21$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 4$$
$$x_{1} x_{2} = -21$$
$$2 x^{2} + 5 x - 25 = 13 x + 17$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - 4 x - 21 = 0$$
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = -4$$
$$q = \frac{c}{a}$$
$$q = -21$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 4$$
$$x_{1} x_{2} = -21$$
Sum and product of roots
[src]
sum
-3 + 7
$$\left(-3\right) + \left(7\right)$$
=
4
$$4$$
product
-3 * 7
$$\left(-3\right) * \left(7\right)$$
=
-21
$$-21$$
The graph