Question

4. Find the zeroes of quadratic polynomial x + 7 + 10.
$x {}^{2} + 7x + 10$
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Answer 1

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Find the zeroes of the Quadratic Polynomial :

x² + 7x + 10 = 0

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★ Concept :

• We can solve these types of questions using two methods -

1. Middle Term Splitting .
2. Using Quadratic Formula .

• Zero of a polynomial - It is that value of the variable , which makes the whole Polynomial equal to zero (0) , it is also called root of the Polynomial .

• Since this Polynomial is Quadratic (2 degree) i.e. highest power of the Polynomial is 2 , so , it will have two roots or zeroes .

★ Solution :

➜ 1st Method -

Splitting the middle term .

$\rm \: {x}^{2} + 7x + 10 = 0$

$\mapsto \rm \: {x}^{2} + 2x + 5x + 10 = 0$

$\mapsto \rm \: x(x + 2) + 5(x + 2) = 0$

$\mapsto \rm \: (x + 5)(x + 2) = 0$

so ,

either

$\sf \: x + 5 = 0 \\ \implies \boxed{\tt \: x = - 5}$

or ,

$\tt \: x + 2 = 0 \\ \implies \boxed{ \tt \: x = - 2}$

So ,

Zeroes : (-5) and (-2)

➜ 2nd Method -

x² + 7x + 10 = 0

here ,

a = 1

b = 7

c = 10

So , using Quadratic Formula ,

$\boxed{ \rm \: roots = \frac{ - b \pm \: \sqrt{ {b}^{2} - 4ac } }{2a} }$

$\mapsto \rm \: roots = \frac{ - 7 \pm \: \sqrt{ {7}^{2} - 4 \times 1 \times 10} }{2 \times 1}$

$\mapsto \rm \: roots = \frac{ - 7 \pm \: \sqrt{49 - 40} }{2}$

$\mapsto \rm \: roots = \frac{ - 7 \pm \: \sqrt{9} }{2}$

$\mapsto \rm \: roots \: = \frac{ - 7 \pm \: 3}{2}$

so ,

$\rm \mapsto \: first \: root = \frac{ - 7 - 3}{2} = \frac{ - 10}{2} \\ \boxed{ \rm \: first \: root = - 5}$

and

$\mapsto \rm \: second \: root = \frac{ - 7 + 3}{2} = \frac{ - 4}{2} \\ \boxed{ \rm \: second \: root = - 2}$

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So ,

Zeroes = -2 and -5

Answer 2

Answer:

To find: zeroes of quadratic polynomial x²+7x+10.

By splitting the middle term:-

x² + 7x + 10 = 0

=> x² + (5+2)x + 10 = 0

=> x² + 5x + 2x + 10 = 0

=> x(x+5) + 2(x+5) = 0

=> (x+2) (x+5) = 0

Either,

(x+2) = 0 or (x+5)= 0

x = -2 or x = -5

Therefore, the zeroes of the polynomial (x²+7x+10) are (-5 )and (-2)

By quadratic formula:-

x²+7x+10=0

Let a=1, b=7and c=10

$\bf{x = - 7 \pm \frac{ \sqrt{49 - 40}}{2}}$

$\bf{x = - 7\pm \frac{ \sqrt{9} }{2}}$

$\bf{x = - 7 \pm \frac{3}{2}}$

Now,

$\bf{x = - 7 - \frac{ 3}{2}} \\ \bf{x = \frac{ - 10}{2}} \\ \bf{x = - 5}$

$\bf{x = - 7 + \frac{3}{2}} \\ \bf{x = \frac{ - 4}{2}} \\ \bf{x = - 2}$

Here too, the zeroes of the polynomial x²+7x+10 is (-5)and (-2)