
How do you solve the equation ${x^2} - 3x - 2 = 0$ using the quadratic formula?
Answer
444.6k+ 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
Master Class 11 Accountancy: Engaging Questions & Answers for Success

Glucose when reduced with HI and red Phosphorus gives class 11 chemistry CBSE

The highest possible oxidation states of Uranium and class 11 chemistry CBSE

Find the value of x if the mode of the following data class 11 maths CBSE

Which of the following can be used in the Friedel Crafts class 11 chemistry CBSE

A sphere of mass 40 kg is attracted by a second sphere class 11 physics CBSE

Trending doubts
10 examples of friction in our daily life

Difference Between Prokaryotic Cells and Eukaryotic Cells

State and prove Bernoullis theorem class 11 physics CBSE

What organs are located on the left side of your body class 11 biology CBSE

Define least count of vernier callipers How do you class 11 physics CBSE

The combining capacity of an element is known as i class 11 chemistry CBSE
