Answer

Verified

483k+ views

Hint: First of all, compare the given equation with a general quadratic equation \[a{{x}^{2}}+bx+c\] and find the values of a, b and c. Then find the discriminant, \[D={{b}^{2}}-4ac\]. If D > 0, then roots are real and distinct. If D = 0, then roots are real and equal, and if D < 0, then roots are imaginary.

Complete step-by-step answer:

Here we have to find the nature of the roots of the equation, \[{{x}^{2}}-5x+7=0\].

Before, proceeding with the question, we must know what the roots, nature of roots, etc are. First of all, roots or zeros

of any equation are the values of the variable which satisfy the equation. In other words, if by substituting some value of the variable, the equation becomes zero, then that value of the variable is the root of the equation. For example, equation 5x + 10 = 0 has a root – 2 because by substituting x = – 2, this equation becomes 0.

The nature of roots tells us whether the roots of the equation are real or imaginary, distinct or coincident, etc.

For any general quadratic equation, \[a{{x}^{2}}+bx+c=0\] where \[a,b,c\in R\] and \[a\ne 0\], then, \[D={{b}^{2}}-4ac\] is known as the discriminant of the quadratic equation.

(i) For D > 0, roots are real and distinct (unequal)

(ii) For D = 0, roots are real and coincident (equal)

(iii), For D < 0, roots are imaginary

Let us consider the quadratic equation given in the question \[{{x}^{2}}-5x+7=0\]

By comparing it to the general quadratic equation \[a{{x}^{2}}+bx+c=0\], we get, a = 1, b = -5 and c = 7

We know that \[D={{b}^{2}}-4ac\].

By substituting the values of a, b and c, we get

\[D={{\left( -5 \right)}^{2}}-4\left( 1 \right)\left( 7 \right)\]

By simplifying RHS of the above equation, we get,

D = 25 – 28

D = – 3

Here, - 3 < 0 that means D < 0.

Hence, we get that the roots of the quadratic equation \[{{x}^{2}}-5x+7=0\] are imaginary in nature.

Note:

Some students make the mistake of writing discriminant D as \[\sqrt{{{b}^{2}}-4ac}\] while D is equal to \[{{b}^{2}}-4ac\]. Also, students should properly take the values of a, b, and c and especially take care of their signs while comparing the given quadratic equation with the general quadratic equation \[a{{x}^{2}}+bx+c\]. Here, \[a,b,c\in R\] and \[a\ne 0\].

Complete step-by-step answer:

Here we have to find the nature of the roots of the equation, \[{{x}^{2}}-5x+7=0\].

Before, proceeding with the question, we must know what the roots, nature of roots, etc are. First of all, roots or zeros

of any equation are the values of the variable which satisfy the equation. In other words, if by substituting some value of the variable, the equation becomes zero, then that value of the variable is the root of the equation. For example, equation 5x + 10 = 0 has a root – 2 because by substituting x = – 2, this equation becomes 0.

The nature of roots tells us whether the roots of the equation are real or imaginary, distinct or coincident, etc.

For any general quadratic equation, \[a{{x}^{2}}+bx+c=0\] where \[a,b,c\in R\] and \[a\ne 0\], then, \[D={{b}^{2}}-4ac\] is known as the discriminant of the quadratic equation.

(i) For D > 0, roots are real and distinct (unequal)

(ii) For D = 0, roots are real and coincident (equal)

(iii), For D < 0, roots are imaginary

Let us consider the quadratic equation given in the question \[{{x}^{2}}-5x+7=0\]

By comparing it to the general quadratic equation \[a{{x}^{2}}+bx+c=0\], we get, a = 1, b = -5 and c = 7

We know that \[D={{b}^{2}}-4ac\].

By substituting the values of a, b and c, we get

\[D={{\left( -5 \right)}^{2}}-4\left( 1 \right)\left( 7 \right)\]

By simplifying RHS of the above equation, we get,

D = 25 – 28

D = – 3

Here, - 3 < 0 that means D < 0.

Hence, we get that the roots of the quadratic equation \[{{x}^{2}}-5x+7=0\] are imaginary in nature.

Note:

Some students make the mistake of writing discriminant D as \[\sqrt{{{b}^{2}}-4ac}\] while D is equal to \[{{b}^{2}}-4ac\]. Also, students should properly take the values of a, b, and c and especially take care of their signs while comparing the given quadratic equation with the general quadratic equation \[a{{x}^{2}}+bx+c\]. Here, \[a,b,c\in R\] and \[a\ne 0\].

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

Which places in India experience sunrise first and class 9 social science CBSE

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

In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Why is there a time difference of about 5 hours between class 10 social science CBSE