Answer
384.9k+ views
Hint: A polynomial of degree two is called a quadratic polynomial and its zeros can be found using many methods like factorization, completing the square, graphs, quadratic formula etc. The quadratic formula is used when we fail to find the factors of the equation. If factors are difficult to find then we use Sridhar’s formula to find the roots. That is \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\].
Complete step-by-step solution:
Given, \[2{x^2} + 8x + 8 = 0\]
Since the degree of the equation is 2, we have 2 factors.
On comparing the given equation with the standard quadratic equation\[a{x^2} + bx + c = 0\], we have\[a = 2\], \[b = 8\] and \[c = 8\].
The standard form of the factorization of quadratic equation is \[a{x^2} + {b_1}x + {b_2}x + c = 0\], which satisfies the condition \[{b_1} \times {b_2} = a \times c\] and \[{b_1} + {b_2} = b\].
We can write the given equation as \[2{x^2} + 4x + 4x + 8 = 0\], where \[{b_1} = 4\] and \[{b_2} = 4\]. Also \[{b_1} \times {b_2} = 4 \times 4 = 16(ac)\] and \[{b_1} + {b_2} = 4 + 4 = 8(b)\].
Thus we have,
\[ \Rightarrow 2{x^2} + 8x + 8 = 2{x^2} + 4x + 4x + 8\]
\[ = 2{x^2} + 4x + 4x + 8\]
Taking ‘2x’ common in the first two terms and taking 4 common in the remaining two terms we have,
\[ = 2x\left( {x + 2} \right) + 4(x + 2)\]
Again taking \[\left( {x + 2} \right)\] common we have,
\[ = \left( {x + 2} \right)\left( {x + 2} \right)\]
Hence the factors of \[2{x^2} - 6x + 4 = 0\] are \[\left( {x + 2} \right)\] and \[\left( {x + 2} \right)\].
(We can also find the roots of the given quadratic equation by equating the obtained factors to zero. That is
\[\left( {x + 2} \right)\left( {x + 2} \right) = 0\]
By zero product principle we have,
\[\left( {x + 2} \right) = 0\] or \[\left( {x + 2} \right) = 0\]
\[x = - 2\] or \[x = - 2\]these are the roots)
Note: We can easily solve this if we know some idea of algebraic identity,
\[2{x^2} + 8x + 8 = 0\]
We can divide the whole equation by 2, to make it simple
\[{x^2} + 4x + 4 = 0\]
But we know \[{(a + b)^2} = {a^2} + 2ab + {b^2}\], applying this we have,
\[{(x + 2)^2} = 0\]
Hence the factors of \[2{x^2} - 6x + 4 = 0\] are \[\left( {x + 2} \right)\] and \[\left( {x + 2} \right)\]
These are the roots of the given polynomial of degree 2. In above, if we are unable to expand the middle term of the given equation into a sum of two numbers then we use a quadratic formula to solve the given problem. Quadratic formula and Sridhar’s formula are both the same. Careful in the calculation part.
Complete step-by-step solution:
Given, \[2{x^2} + 8x + 8 = 0\]
Since the degree of the equation is 2, we have 2 factors.
On comparing the given equation with the standard quadratic equation\[a{x^2} + bx + c = 0\], we have\[a = 2\], \[b = 8\] and \[c = 8\].
The standard form of the factorization of quadratic equation is \[a{x^2} + {b_1}x + {b_2}x + c = 0\], which satisfies the condition \[{b_1} \times {b_2} = a \times c\] and \[{b_1} + {b_2} = b\].
We can write the given equation as \[2{x^2} + 4x + 4x + 8 = 0\], where \[{b_1} = 4\] and \[{b_2} = 4\]. Also \[{b_1} \times {b_2} = 4 \times 4 = 16(ac)\] and \[{b_1} + {b_2} = 4 + 4 = 8(b)\].
Thus we have,
\[ \Rightarrow 2{x^2} + 8x + 8 = 2{x^2} + 4x + 4x + 8\]
\[ = 2{x^2} + 4x + 4x + 8\]
Taking ‘2x’ common in the first two terms and taking 4 common in the remaining two terms we have,
\[ = 2x\left( {x + 2} \right) + 4(x + 2)\]
Again taking \[\left( {x + 2} \right)\] common we have,
\[ = \left( {x + 2} \right)\left( {x + 2} \right)\]
Hence the factors of \[2{x^2} - 6x + 4 = 0\] are \[\left( {x + 2} \right)\] and \[\left( {x + 2} \right)\].
(We can also find the roots of the given quadratic equation by equating the obtained factors to zero. That is
\[\left( {x + 2} \right)\left( {x + 2} \right) = 0\]
By zero product principle we have,
\[\left( {x + 2} \right) = 0\] or \[\left( {x + 2} \right) = 0\]
\[x = - 2\] or \[x = - 2\]these are the roots)
Note: We can easily solve this if we know some idea of algebraic identity,
\[2{x^2} + 8x + 8 = 0\]
We can divide the whole equation by 2, to make it simple
\[{x^2} + 4x + 4 = 0\]
But we know \[{(a + b)^2} = {a^2} + 2ab + {b^2}\], applying this we have,
\[{(x + 2)^2} = 0\]
Hence the factors of \[2{x^2} - 6x + 4 = 0\] are \[\left( {x + 2} \right)\] and \[\left( {x + 2} \right)\]
These are the roots of the given polynomial of degree 2. In above, if we are unable to expand the middle term of the given equation into a sum of two numbers then we use a quadratic formula to solve the given problem. Quadratic formula and Sridhar’s formula are both the same. Careful in the calculation part.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)