Question

Find the zeroes of the quadratic polynomial and verify the relationship between the zeroes and the coefficients(i) x2 + 4x -5 (ii) 2x2 -8x +6

Answer 1

SolutioN :

Let,

• Both Zeros of polynomial be α and β.

( i ) x² + 4x - 5.

Compare with General Equation.

$\tt \dagger \: \: \: \: \: \fbox{x = \dfrac{ - b \pm \sqrt{ {b}^{2} - 4ac } }{2a} }$

Where as,

• a = 1.
• b = 4.
• c = - 5.

☛ Let Find Zeros.

$\tt : \implies x = \dfrac{ - b \pm \sqrt{ {b}^{2} - 4ac } }{2a}$

$\tt : \implies x = \dfrac{ - 4 \pm \sqrt{ {4}^{2} - 4 \times 1 \times - 5} }{2}$

$\tt : \implies x = \dfrac{ - 4 \pm \sqrt{ 16 + 20} }{2}$

$\tt : \implies x = \dfrac{ - 4 \pm \sqrt{ 36} }{2}$

$\tt : \implies x = \dfrac{ - 4 \pm 6 }{2}$

Either,

$\tt : \implies x = \dfrac{ - 4 + 6}{2}$

$\tt : \implies x = \dfrac{ 2}{2}$

$\tt : \implies x = 1.$

Or,

$\tt : \implies x = \dfrac{ - 4 - 6}{2}$

$\tt : \implies x = - 5.$

$\rule{120}3$

☛ Let Verify Zeros and Coefficient.

• α = 1.
• β = - 5.

Now,

☛ Sum of Zeros.

➟ α + β = - b / a.

➟ 1 - 5 = - 4.

$\rule{90}1$

☛ Product of Zeros.

➟ α * β = c / a.

➟ 1 * - 5 = - 5.

➟ - 5 = - 5.

★ Hence Verify.

Note : part ( ii ) provide above.