If the zeroes of the polynomial x² + px + q are double in value to the zeroes of 2x² - 5x - 3, find the value of and .

Answer:

The values of "p" and "q" are p = -5 and q = -6

Step-by-step explanation:

Let's start by finding the zeroes of the polynomial 2x² - 5x - 3, and then we'll determine the relationship between these zeroes and the zeroes of x² + px + q.

The zeroes of a quadratic polynomial of the form ax² + bx + c can be found using the quadratic formula:

For the polynomial 2x² - 5x - 3, a = 2, b = -5, and c = -3. So, the quadratic formula becomes:

x = [-b ± √(b² - 4ac)] / (2a)

Substitute the values:

x = [-(-5) ± √((-5)² - 4(2)(-3))] / (2(2))

Simplify:

x = (5 ± √(25 + 24)) / 4

x = (5 ± √49) / 4

x = (5 ± 7) / 4

Now, we have two possible values for x:

x₁ = (5 + 7) / 4 = 12/4 = 3

x₂ = (5 - 7) / 4 = -2/4 = -1/2

So, the zeroes of 2x² - 5x - 3 are x₁ = 3 and x₂ = -1/2.

Now, we need to find the relationship between these zeroes and the zeroes of x² + px + q.

If the zeroes of x² + px + q are double in value to the zeroes of 2x² - 5x - 3, it means that for each zero "x" of 2x² - 5x - 3, there will be a corresponding zero "2x" for x² + px + q.

So, for x² + px + q, the zeroes will be 2 times the zeroes of 2x² - 5x - 3:

For x₁ = 3, the corresponding zero for x² + px + q is 2x₁ = 2(3) = 6.

For x₂ = -1/2, the corresponding zero for x² + px + q is 2x₂ = 2(-1/2) = -1.

Now, we have the zeroes of x² + px + q: 6 and -1.

To find "p" and "q," we can use Vieta's formulas. Vieta's formulas state that for a quadratic polynomial of the form ax² + bx + c with zeroes α and β:

α + β = -b/a

α * β = c/a

In our case, for x² + px + q with zeroes 6 and -1:

α + β = 6 - 1 = 5

α * β = 6 * (-1) = -6

Now, let's match these with the coefficients of x² + px + q:

α + β = 5, which corresponds to -p (since there's an "x" term in the middle)

α * β = -6, which corresponds to q (the constant term)

So, we have the following equations:

-p = 5

q = -6

Solve for "p" and "q":

p = -5

q = -6

So, the values of "p" and "q" are p = -5 and q = -6.

If the zeroes of the polynomial x² + px + q are double in value to the zeroes of 2x² - 5x - 3, find the value of p and q

Answer:

 p  and  q  are -5 and -6 respectively.

Step-by-step explanation:

factor

2x²-5x-3=0

(x-3) (2x + 1) = 0

x = 3, -1/2

multiply both by 2 = "double in value to the zeroes"

x = 6, -1

reverse factor them

(x-6)(x+1)

multiply

x2−5x−6

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