How many positive two-digit integers are factors of both 100 and 150?

Mathematics

1 answer:

Nataliya [291]3 years ago

3 0

Answer:

There are 3 positive 2 digit integers that are factors of both 100 and 150

Step-by-step explanation:

Factors of 100:

1, 2, 4, 5, 10, 20, 25, 50, 100

Factors of 150:

1, 2, 4, 5, 6, 10, 15, 25, 30, 50, 75, 150

Common factors between these numbers:

1, 2, 4, 5, 10, 25, 50

Common factors that are positive 2 digit integers:

10, 25, 50

There are 3 positive 2 digit integers that are factors of both 100 and 150

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notka56 [123]

Answer:

x = -3, 0, or 7

Step-by-step explanation:

After removing common factors, the remaining quadratic can be factored by comparison to the factored form of a quadratic.

Write the equation in standard form.

4x³ -16x² -84x = 0

Factor out the common factor from all terms.

= 4x(x² -4x -21) = 0

Compare to the factored form of a quadratic:

(x +a)(x +b) = x² +(a+b)x +ab

This tells you the constants 'a' and 'b' in the factors can be found by considering ...

(a+b) = -4 . . . . the coefficient of the x term of the quadratic

ab = -21 . . . . . the constant term of the quadratic

It is often helpful to list factor pairs of the constant:

-21 = (-21)(1) = (-7)(3) . . . . integer pairs that have a negative sum

The sums of these pairs are -20 and -4. We are interested in the latter. We can choose ...

a = -7, b = 3

Put it all together.

4x³ -16x² -84 = (4x)(x -7)(x +3) = 0 . . . . . factored form of the equation

Apply the zero product rule. This rule tells you the product of factors will be zero when one or more of the factors is zero:

4x = 0 ⇒ x = 0

x -7 = 0 ⇒ x = 7

x +3 = 0 ⇒ x = -3

Solutions to the equation are x ∈ {-3, 0, 7}.

_____

<em>Additional comment</em>

What we did in Step 3 is sometimes referred to as the X-method of factoring a quadratic. The constant (ab product) is put at the top of the X, and the sum (a+b) is put at the bottom. The sides of the X are filled in with values that match the product and sum: -7 and 3. The method is modified slightly if the coefficient of x² is not 1.

A graphing calculator often provides a quick and easy method of finding the real zeros of a polynomial.