Question

One of the roots of the equation 2x^2−bx−20=0 is −2.5. Find the other root

Answer 1

The "other" or "second" root is 4.

Explanation:

We are told that -2.5 is a root of the equation. The coefficient b of the x term is unknown, and must be determined. Because -2.5 is a root, synthetic division with -2.5 as divisor must return a remainder of zero.

Setting up synthetic division, we arrive at:

-2.5 / 2 -b -20

-5 +12.5 + 2.5b

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2 -5-b -7.5 + 2.5b

The remainer, -7.5 + 2.5b, must be zero (0). Thus, 2.5b = 7.5, and b = 3.

Then the other factor has the coefficients {2, -5-b}, and because b = 3, this comes out to coefficients {2, -8}.

The other factor is 2x - 8, which, if set equal to 0, yields x = 4. This is the "other root."

Answer 2

The answer to your question is x = 4

Explanation:

2x² - bx - 20 = 0

One root is -2.5

Process

Get the value of the equation when x = -2.5

2(-2.5)² - b(-2.5) - 20 = 0

2(6.25) + 2.5b - 20 = 0

12.5 + 2.5b - 20 = 0

2.5b = 20 - 12.5

2.5b = 7.5

b = 7.5 / 2.5

b = 3

Then

2x² - 3x - 20 = 0

Factor the polynomial

2 x -20 = -40

2x² -8x + 5x - 20 = 0

2x(x - 4) + 5(x - 4) = 0

(x - 4)(2x + 5) = 0

x₁ - 4 = 0 2x₂ + 5 = 0

x₁ = 4 x₂ = -5/2

x₂ = -2.5