Answer
Verified
415.2k+ views
Hint: This question is from the topic of solutions of quadratic equations. We have to solve this question using the quadratic formula. To solve this we need to know the quadratic formula and discriminant of a quadratic equation. Discriminant of a quadratic equation gives details about the nature of the roots of a quadratic equation. This question is very easy. You just need to apply the formula. Try once by yourself before looking at a complete solution.
Complete step by step solution:
Let us try to solve this question in which we have to find the roots of a quadratic equation ${x^2} - 3x - 2 = 0$ using the quadratic formula. Quadratic formula is given by $\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$ for any general quadratic equation $a{x^2} + bx + c = 0$ where ${b^2} - 4ac$ is called the discriminant of quadratic equation, it tells the nature of roots of quadratic equation. Here are conditions:
1) Two distinct real roots, if ${b^2} - 4ac > 0$
2) Two equal real roots, if ${b^2} - 4ac = 0$
3) No real roots, if ${b^2} - 4ac < 0$
In the given quadratic equation we have,
$
a = 1 \\
b = - 3 \\
c = - 2 \\
$
Discriminant of the quadratic equation is
$
{b^2} - 4ac = {(3)^2} - 4 \cdot 1 \cdot ( - 2) \\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 9 + 8 \\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 17 > 0 \\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \\
$
Hence the given quadratic equation has two distinct real roots, because discriminant is greater than $0$.
Now putting values of $a,b$ and $c$ in quadratic formula we get,
\[
x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} = \dfrac{{ - ( - 3) \pm \sqrt {{{( - 3)}^2} - 4 \cdot 1 \cdot ( -
2)} }}{2} \\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \dfrac{{3 \pm \sqrt {9 + 8} }}{2} \\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \dfrac{{3 \pm \sqrt {17} }}{2} \\
\]
Hence the root of quadratic equation ${x^2} - 3x - 2 = 0$ are $x = \dfrac{{3 + \sqrt {17} }}{2}$and$x = \dfrac{{3 - \sqrt {17} }}{2}$.
Note: To solve questions in which you are asked to find the roots of quadratic equations by quadratic formula you must need to know the formula. We can also solve this by using other methods of finding roots of a quadratic equation such as completing square method and factor method.
Complete step by step solution:
Let us try to solve this question in which we have to find the roots of a quadratic equation ${x^2} - 3x - 2 = 0$ using the quadratic formula. Quadratic formula is given by $\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$ for any general quadratic equation $a{x^2} + bx + c = 0$ where ${b^2} - 4ac$ is called the discriminant of quadratic equation, it tells the nature of roots of quadratic equation. Here are conditions:
1) Two distinct real roots, if ${b^2} - 4ac > 0$
2) Two equal real roots, if ${b^2} - 4ac = 0$
3) No real roots, if ${b^2} - 4ac < 0$
In the given quadratic equation we have,
$
a = 1 \\
b = - 3 \\
c = - 2 \\
$
Discriminant of the quadratic equation is
$
{b^2} - 4ac = {(3)^2} - 4 \cdot 1 \cdot ( - 2) \\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 9 + 8 \\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 17 > 0 \\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \\
$
Hence the given quadratic equation has two distinct real roots, because discriminant is greater than $0$.
Now putting values of $a,b$ and $c$ in quadratic formula we get,
\[
x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} = \dfrac{{ - ( - 3) \pm \sqrt {{{( - 3)}^2} - 4 \cdot 1 \cdot ( -
2)} }}{2} \\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \dfrac{{3 \pm \sqrt {9 + 8} }}{2} \\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \dfrac{{3 \pm \sqrt {17} }}{2} \\
\]
Hence the root of quadratic equation ${x^2} - 3x - 2 = 0$ are $x = \dfrac{{3 + \sqrt {17} }}{2}$and$x = \dfrac{{3 - \sqrt {17} }}{2}$.
Note: To solve questions in which you are asked to find the roots of quadratic equations by quadratic formula you must need to know the formula. We can also solve this by using other methods of finding roots of a quadratic equation such as completing square method and factor method.
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 are the Top 10 Largest Countries of the World?
Who was the Governor general of India at the time of class 11 social science 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
One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE
Difference Between Plant Cell and Animal Cell