Question

If alpha and beta are the zeros of a quadratic polynomial 2 X square + 3 x minus 6 then find the value of Alpha square plus beta square minus Alpha into beta

Answer 1

$\blue{\bold{\underline{\underline{Given}}}}$

• α and β are the zeros of a quadratic polynomial 2x² + 3x - 6

$\blue{\bold{\underline{\underline{ToFind}}}}$

• Value of α² + β² - αβ

$\blue{\bold{\underline{\underline{Solution}}}}$

Now,

On comparing 2x² + 3x - 6 = 0with standard form ax² + bx + c = 0,

We get,

➮ a = 2

➮ b = 3

➮ c = -6

Now,

We know that,

➡ α + β = -b/a = -3/2. ---(1)

➡ αβ = c/a = -6/2 = -3. --- (2)

Now,

↣ (α + β) = -3/2

On squaring both sides, we get

↣ (α + β) ² = (-3/2)²

↣ α² + β² + 2αβ = 9/4

↣ α² + β² + 2×-3 = 9/4. --(from(2)

↣ α² + β² - 6 = 9/4

↣ α² + β² = 9/4 + 6

↣ α² + β² = 9/4 + 6×4/1×4

↣ α² + β² = 33/4. ---(3)

Now,

↣ α² + β² - αβ

↣ 33/4 - (-3) ----{from (2) and (3)}

↣ 33/4 + 3

↣ 45/4

Answer 2

Therefore the value of α² + β² - αβ is '33/4'.

Given:

The quadratic equation: 2x² + 3x -6 = 0

To Find:

The value of α² + β² - αβ

Solution:

The given question can be solved very easily as shown below.

Concept:

If a quadratic equation is given as ax² + bx + c = 0 and 'α' and 'β' are the roots of the given quadratic equation.

Then the sum of roots = α + β = -b/a

And Product of roots = αβ = c/a

Given quadratic equation: 2x² + 3x -6 = 0

In comparing to the above equation,

a = 2, b = 3, c = -6

Then the sum of the roots = ( α + β ) = -b/a = -3/2

Product of the roots = αβ = -6/2 = -3

⇒ The value to be found is α² + β² - αβ

Adding and subtracting αβ in the above equation,

⇒ α² + β² - αβ + αβ - αβ = α² + β² + αβ - αβ - αβ

⇒ ( α + β )² - 2αβ

Now substituting the values,

⇒ ( α + β )² - 2αβ = ( -3/2 )² - 2( -3 ) = 9/4 + 6 = 33/4

Therefore the value of α² + β² - αβ is '33/4'.

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