Answer
398.7k+ views
Hint: We can solve this using algebraic identities. We use the identity \[{a^2} - {b^2} = (a - b)(a + b)\] to solve the given problem. We can see that 49 and 36 are perfect squares. We can convert the given problem into \[{a^2} - {b^2}\], since the square of 3 is 9 and square of 2 is 4.
Complete step-by-step solution:
Given, \[9{x^2} - 4\]
We can rewrite it as \[ = {3^2}.{x^2} - {2^2}\]
\[ = {(3x)^2} - {2^2}\].
That is it is in the form \[{a^2} - {b^2}\], where \[a = 3x\] and \[b = 2\].
We have the formula \[{a^2} - {b^2} = (a - b)(a + b)\].
Then above becomes,
\[ = (3x - 2)(3x + 2)\]. These are the factors of the \[9{x^2} - 4\].
Additional Information:
We can find the root of the polynomial by equating the obtained factors to zero. That is
\[ \Rightarrow (3x - 2)(3x + 2) = 0\]
Using the zero product principle, that is of ab=0 then a=0 or b=0.using this we get,
\[ \Rightarrow 3x - 2 = 0\] and \[3x + 2 = 0\].
\[ \Rightarrow 3x = 2\] and \[3x = - 2\].
\[ \Rightarrow x = \dfrac{2}{3}\] and \[x = - \dfrac{2}{3}\]. This is the roots of the given polynomial.
We know that on the x-axis, the value of y is zero so the roots of an equation are the points on the x-axis that is the roots are simply the x- intercept.
Note: Follow the same procedure for these kinds of problems. Since the given equation is a polynomial. The highest exponent of the polynomial in a polynomial equation is called its degree. A polynomial equation has exactly as many roots as its degree. Here the degree is 2. Hence it is called a quadratic equation. (We know the quadratic equation is of the form \[a{x^2} + bx + c = 0\], in our problem coefficient of ‘x’ is zero) Hence, we have two roots or two factors.
Complete step-by-step solution:
Given, \[9{x^2} - 4\]
We can rewrite it as \[ = {3^2}.{x^2} - {2^2}\]
\[ = {(3x)^2} - {2^2}\].
That is it is in the form \[{a^2} - {b^2}\], where \[a = 3x\] and \[b = 2\].
We have the formula \[{a^2} - {b^2} = (a - b)(a + b)\].
Then above becomes,
\[ = (3x - 2)(3x + 2)\]. These are the factors of the \[9{x^2} - 4\].
Additional Information:
We can find the root of the polynomial by equating the obtained factors to zero. That is
\[ \Rightarrow (3x - 2)(3x + 2) = 0\]
Using the zero product principle, that is of ab=0 then a=0 or b=0.using this we get,
\[ \Rightarrow 3x - 2 = 0\] and \[3x + 2 = 0\].
\[ \Rightarrow 3x = 2\] and \[3x = - 2\].
\[ \Rightarrow x = \dfrac{2}{3}\] and \[x = - \dfrac{2}{3}\]. This is the roots of the given polynomial.
We know that on the x-axis, the value of y is zero so the roots of an equation are the points on the x-axis that is the roots are simply the x- intercept.
Note: Follow the same procedure for these kinds of problems. Since the given equation is a polynomial. The highest exponent of the polynomial in a polynomial equation is called its degree. A polynomial equation has exactly as many roots as its degree. Here the degree is 2. Hence it is called a quadratic equation. (We know the quadratic equation is of the form \[a{x^2} + bx + c = 0\], in our problem coefficient of ‘x’ is zero) Hence, we have two roots or two factors.
Recently Updated Pages
Mark and label the given geoinformation on the outline class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
When people say No pun intended what does that mea class 8 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Name the states which share their boundary with Indias class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give an account of the Northern Plains of India class 9 social science 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)
Advantages and disadvantages of science
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Bimbisara was the founder of dynasty A Nanda B Haryanka class 6 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)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
10 examples of evaporation in daily life with explanations
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![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)
How do you graph the function fx 4x class 9 maths 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)