Show that the equation x^3 + x - 3 has a solution between 1 and 2

see explanation

Step-by-step explanation:

Evaluate the polynomial at x = 1 and x = 2

x = 1 : 1³ + 1 - 3 = 1 + 1 - 3 = - 1

x = 2 : 2³ + 2 - 3 = 8 + 2 - 3 = 7

Then ( 1, - 1 ) → (2, 7 )

Shows the polynomial goes from y = - 1 to y = + 7

The change in sign indicates the graph crosses the x- axis between 1 and 2

Thus indicating a solution between x = 1 and x = 2

step-by-step explanation:

equation

you are asked to write the equation

  expression = 0

using the given expression. when you do that, you have

  2x^2-2x-12 = 0

factoring

you can see that each of the coefficients is even, so a factor of 2 can be removed immediately.

  2(x^2 -x -6) = 0

if you multiply the binomials (x+a)(x+b), you get

  (x +a)(x +b) = x^2 +(a+b)x +ab

this tells you the coefficient of x is the sum of two factors of the constant term. here, that means -1 is the sum of two factors of -6. we want to know those factors so we can factor the trinomial.

  -6 = -1×6 = -2×3 = -3×2 = -6×1

the factor pair that sums to -1 is {-3, 2}, so our factorization is

  2(x -3)(x +2) = 0

solutions

the solutions to this equation are the values of x that make the product zero. a product is zero only when only of the factors of the product is zero. here, that means

  x - 3 = 0

  x = 3 . . . add 3 to both sides of the equation

or

  x +2 = 0

  x = -2 . . . add -2 to both sides of the equation

the solutions to the equation are x=3 or x=-2. both of these values of x satisfy the equation—they make it be true.

step-by-step explanation: