Answer
Verified
427.8k+ views
Hint: The general form of a quadratic equation is \[a{x^2} + bx + c = 0\], where $ a \ne 0 $ . The values of $ x $ which satisfy the equation are called the roots or the solutions. These roots can be real or imaginary. There are two ways to find solutions/roots of $ x $ – comparing method and discriminant method.
Complete step-by-step answer:
Under comparing methods, we keep terms both on the left-hand and right-hand side and keep on comparing them until and unless the left-hand side is equal to the right-hand side. Let us try to solve the given question by comparing methods.
First, we have to add $ 5 $ to both the sides of the equation.
$
\Rightarrow 2{x^2} - 5 + 5 = 0 + 5 \\
\Rightarrow 2{x^2} = 5 \;
$
Next, we are supposed to divide both the sides of the equation by $ 2 $ .
$
\Rightarrow \dfrac{{2{x^2}}}{2} = \dfrac{5}{2} \\
\Rightarrow {x^2} = \dfrac{5}{2} \;
$
Now, taking the square root of both sides of the equation in order to solve for $ x $ .
$
\Rightarrow \sqrt {{x^2}} = \sqrt {\dfrac{5}{2}} \\
\Rightarrow x = \pm \dfrac{{\sqrt 5 }}{{\sqrt 2 }} \;
$
We can further simplify this solution by rationalizing the denominator. In order to do that, we have to multiply and divide the solution with $ \sqrt 2 $ .
$
\Rightarrow x = \pm \dfrac{{\sqrt 5 \times \sqrt 2 }}{{\sqrt 2 \times \sqrt 2 }} \\
\Rightarrow x = \pm \dfrac{{\sqrt {10} }}{2} \;
$
This gives us the two required solutions and that are $ x = \dfrac{{\sqrt {10} }}{2} $ and $ x = - \dfrac{{\sqrt {10} }}{2} $ .
So, the correct answer is “ $ x = \dfrac{{\sqrt {10} }}{2} $ and $ x = - \dfrac{{\sqrt {10} }}{2} $ ”.
Note: Let us try to solve this question by discriminant method. As we know that the quadratic formula is $ x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} $ . Here, $ a = 2,b = 0,c = - 5 $ .
Now by putting values of $ a,b $ and $ c $ in the quadratic formula we get,
$ \Rightarrow x = \dfrac{{ - 0 \pm \sqrt {{0^2} - 4 \times 2 \times \left( { - 5} \right)} }}{{2 \times 2}} $
After simplifying, we get,
$
\Rightarrow x = \dfrac{{0 \pm \sqrt {0 + 40} }}{4} \\
\Rightarrow x = \dfrac{{ \pm \sqrt {40} }}{4} \\
\Rightarrow x = \dfrac{{ \pm 2\sqrt {10} }}{4} \\
\Rightarrow x = \dfrac{{ \pm \sqrt {10} }}{2} \;
$
Hence, the two values of $ x $ are $ x = \dfrac{{\sqrt {10} }}{2} $ and $ x = - \dfrac{{\sqrt {10} }}{2} $ .
There are other ways to solve quadratic equations such as factorization or completing the square. These methods/ways couldn’t be used in the given question because the value of $ x $ was missing. In order to solve questions by these methods, the question should be in standard form i.e., $ a{x^2} + bx + c = 0 $ where $ a \ne 0 $ and also $ b \ne 0 $ . Every quadratic equation has two solutions.
Complete step-by-step answer:
Under comparing methods, we keep terms both on the left-hand and right-hand side and keep on comparing them until and unless the left-hand side is equal to the right-hand side. Let us try to solve the given question by comparing methods.
First, we have to add $ 5 $ to both the sides of the equation.
$
\Rightarrow 2{x^2} - 5 + 5 = 0 + 5 \\
\Rightarrow 2{x^2} = 5 \;
$
Next, we are supposed to divide both the sides of the equation by $ 2 $ .
$
\Rightarrow \dfrac{{2{x^2}}}{2} = \dfrac{5}{2} \\
\Rightarrow {x^2} = \dfrac{5}{2} \;
$
Now, taking the square root of both sides of the equation in order to solve for $ x $ .
$
\Rightarrow \sqrt {{x^2}} = \sqrt {\dfrac{5}{2}} \\
\Rightarrow x = \pm \dfrac{{\sqrt 5 }}{{\sqrt 2 }} \;
$
We can further simplify this solution by rationalizing the denominator. In order to do that, we have to multiply and divide the solution with $ \sqrt 2 $ .
$
\Rightarrow x = \pm \dfrac{{\sqrt 5 \times \sqrt 2 }}{{\sqrt 2 \times \sqrt 2 }} \\
\Rightarrow x = \pm \dfrac{{\sqrt {10} }}{2} \;
$
This gives us the two required solutions and that are $ x = \dfrac{{\sqrt {10} }}{2} $ and $ x = - \dfrac{{\sqrt {10} }}{2} $ .
So, the correct answer is “ $ x = \dfrac{{\sqrt {10} }}{2} $ and $ x = - \dfrac{{\sqrt {10} }}{2} $ ”.
Note: Let us try to solve this question by discriminant method. As we know that the quadratic formula is $ x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} $ . Here, $ a = 2,b = 0,c = - 5 $ .
Now by putting values of $ a,b $ and $ c $ in the quadratic formula we get,
$ \Rightarrow x = \dfrac{{ - 0 \pm \sqrt {{0^2} - 4 \times 2 \times \left( { - 5} \right)} }}{{2 \times 2}} $
After simplifying, we get,
$
\Rightarrow x = \dfrac{{0 \pm \sqrt {0 + 40} }}{4} \\
\Rightarrow x = \dfrac{{ \pm \sqrt {40} }}{4} \\
\Rightarrow x = \dfrac{{ \pm 2\sqrt {10} }}{4} \\
\Rightarrow x = \dfrac{{ \pm \sqrt {10} }}{2} \;
$
Hence, the two values of $ x $ are $ x = \dfrac{{\sqrt {10} }}{2} $ and $ x = - \dfrac{{\sqrt {10} }}{2} $ .
There are other ways to solve quadratic equations such as factorization or completing the square. These methods/ways couldn’t be used in the given question because the value of $ x $ was missing. In order to solve questions by these methods, the question should be in standard form i.e., $ a{x^2} + bx + c = 0 $ where $ a \ne 0 $ and also $ b \ne 0 $ . Every quadratic equation has two solutions.
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
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Which are the Top 10 Largest Countries of the World?
Change the following sentences into negative and interrogative class 10 english CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
10 examples of friction in our daily life
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
What is pollution? How many types of pollution? Define it