Question

if a = -1/2 is a root of the quadratic equation 8x²-bx-3 . find the value of b, the other root, and (1/a - 1/b)²​

Answer 1

If a = -1/2 is a root of the quadratic equation 8x² - bx - 3, then we know that when x = -1/2, the equation is equal to 0. We can use this information to solve for b.

Substituting x = -1/2 into the equation, we get:

8(-1/2)² - b(-1/2) - 3 = 0

Simplifying and solving for b, we get:

2 - (b/2) - 3 = 0

b/2 = -1

b = -2

Therefore, b = -2 is the value we are looking for.

To find the other root, we can use the fact that the product of the roots of a quadratic equation is equal to the constant term divided by the leading coefficient. In this case, the constant term is -3 and the leading coefficient is 8. Therefore, the product of the roots is:

(-1/2) times the other root = -3/8

Solving for the other root, we get:

(-1/2) times the other root = -3/8

other root = (-3/8) / (-1/2)

other root = (3/8) * 2

other root = 3/4

Therefore, the other root is 3/4.

Finally, to find (1/a - 1/b)², we can substitute a = -1/2 and b = -2 into the expression:

(1/a - 1/b)² = (1/(-1/2) - 1/(-2))²

= (-2 - 1/2)²

= (-5/2)²

= 25/4

Therefore, (1/a - 1/b)² is equal to 25/4.

Answer 2

`b=2`

`\textsf{Other root} = (3)/(4)`

`\left((1)/(a)-(1)/(b)\right)^2=(25)/(4)`

Explanation:

Roots are also called x-intercepts or zeros. They are the x-values of the points at which the function crosses the x-axis, so the values of x when f(x) = 0.

If x = α is a root of a polynomial f(x), then f(α) = 0.

Therefore, given that a = -1/2 is a root of the quadratic equation 8x² - bx - 3, substitute x = -1/2 into the equation and set it to zero:

`\implies 8\left(-(1)/(2)\right)^2-b\left(-(1)/(2)\right)-3=0`

Solve for b:

`\implies 8\left((1)/(4)\right)+(1)/(2)b-3=0`

`\implies (8)/(4)+(1)/(2)b-3=0`

`\implies 2+(1)/(2)b-3=0`

`\implies (1)/(2)b-1=0`

`\implies (1)/(2)b=1`

`\implies b=2`

Therefore, the quadratic equation is:

`\boxed{8x^2-2x-3}`

The product of the roots of a quadratic equation is equal to the constant term divided by the leading coefficient.

The constant term of the quadratic equation is -3 and the leading coefficient is 8. Let the other root be "r". Therefore:

`\implies a \cdot r=(-3)/(8)`

Substitute the known value of a = -1/2 and solve for r:

`\implies -(1)/(2) \cdot r=(-3)/(8)`

`\implies r=(3)/(4)`

Therefore, the other root of the quadratic equation is 3/4.

To find the value of (1/a - 1/b)²​, substitute the given value of a and the found value of b into the equation and solve:

`\implies \left((1)/(a)-(1)/(b)\right)^2`

`\implies \left((1)/(-(1)/(2))-(1)/(2)\right)^2`

`\implies \left(-2-(1)/(2)\right)^2`

`\implies \left(-(5)/(2)\right)^2`

`\implies (25)/(4)`