Answer
Verified
425.7k+ views
Hint: In this question, we are given a polynomial equation and we have to solve it, that is, we have to find its factors. The degree of a polynomial equation is defined as the highest exponent of the unknown quantity in a polynomial equation. The factors/solution/zeros of the given polynomial are defined as those values of x at which the value of a polynomial is zero. The equation given in the question has a degree 2, so we have a quadratic equation and we are already told to solve the given quadratic equation using the quadratic formula, so for that, we will first express the given equation in the standard equation form and then the values of the coefficients are plugged in the quadratic formula.
Complete step by step solution:
The equation given is $2{x^2} + 9x + 4 = 0$
On comparing the given equation with the standard quadratic equation $a{x^2} + bx + c = 0$ , we get –
$a = 2,\,b = 9,\,c = 4$
The Quadratic formula is given as –
$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Now, put the known values in the above equation –
$
x = \dfrac{{ - 9 \pm \sqrt {{{(9)}^2} - 4(2)(4)} }}{{2(2)}} \\
\Rightarrow x = \dfrac{{ - 9 \pm \sqrt {81 - 32} }}{4} \\
\Rightarrow x = \dfrac{{ - 9 \pm \sqrt {49} }}{4} \\
\Rightarrow x = \dfrac{{ - 9 \pm 7}}{4} \\
\Rightarrow x = \dfrac{{ - 9 + 7}}{4},\,x = \dfrac{{ - 9 - 7}}{4} \\
\Rightarrow x = - \dfrac{1}{2},\,x = - 4 \\
$
Hence the zeros of the given equation are $ - \dfrac{1}{2}$ and $ - 4$ .
Note: When the alphabets representing an unknown variable quantity in an algebraic expression are raised to some non-negative integer as the power, the algebraic expression becomes a polynomial equation. A quadratic polynomial is defined as a polynomial of degree two and its zeros can be found using many methods like factorization, completing the square, graphs, quadratic formula etc. We use the quadratic formula when we fail to find the factors of the equation. But in this question we can also solve the equation by factorization as the factors of the equation can be found easily by hit and trial.
Complete step by step solution:
The equation given is $2{x^2} + 9x + 4 = 0$
On comparing the given equation with the standard quadratic equation $a{x^2} + bx + c = 0$ , we get –
$a = 2,\,b = 9,\,c = 4$
The Quadratic formula is given as –
$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Now, put the known values in the above equation –
$
x = \dfrac{{ - 9 \pm \sqrt {{{(9)}^2} - 4(2)(4)} }}{{2(2)}} \\
\Rightarrow x = \dfrac{{ - 9 \pm \sqrt {81 - 32} }}{4} \\
\Rightarrow x = \dfrac{{ - 9 \pm \sqrt {49} }}{4} \\
\Rightarrow x = \dfrac{{ - 9 \pm 7}}{4} \\
\Rightarrow x = \dfrac{{ - 9 + 7}}{4},\,x = \dfrac{{ - 9 - 7}}{4} \\
\Rightarrow x = - \dfrac{1}{2},\,x = - 4 \\
$
Hence the zeros of the given equation are $ - \dfrac{1}{2}$ and $ - 4$ .
Note: When the alphabets representing an unknown variable quantity in an algebraic expression are raised to some non-negative integer as the power, the algebraic expression becomes a polynomial equation. A quadratic polynomial is defined as a polynomial of degree two and its zeros can be found using many methods like factorization, completing the square, graphs, quadratic formula etc. We use the quadratic formula when we fail to find the factors of the equation. But in this question we can also solve the equation by factorization as the factors of the equation can be found easily by hit and trial.
Recently Updated Pages
Identify the feminine gender noun from the given sentence class 10 english CBSE
Your club organized a blood donation camp in your city class 10 english CBSE
Choose the correct meaning of the idiomphrase from class 10 english CBSE
Identify the neuter gender noun from the given sentence class 10 english CBSE
Choose the word which best expresses the meaning of class 10 english CBSE
Choose the word which is closest to the opposite in class 10 english CBSE
Trending doubts
Which are the Top 10 Largest Countries of the World?
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
How do you graph the function fx 4x class 9 maths CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
What organs are located on the left side of your body class 11 biology CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Change the following sentences into negative and interrogative class 10 english CBSE
How much time does it take to bleed after eating p class 12 biology CBSE