Example Of Quadratic Equation Whose Nature Of The Roots Is Imaginary

Find the discriminant of the quadratic equation 2 x2 4x 3 0 and hence find the nature of its roots. Hence the roots are Imaginary.

Nature Of The Roots Of A Quadratic Equation

To examine the roots of a quadratic equation let us consider the general form a quadratic equation.

Example of quadratic equation whose nature of the roots is imaginary. X2 4x 5 0. X x - 2 4 upon multiplying and moving the 4 becomes x² - 2x - 4 0 x 2x 3 12 upon multiplying and moving the 12 becomes 2x² - 3x - 12 0 3x x 8 -2 upon multiplying and moving the -2 becomes 3x² 24x 2 0 5x² 9 - x moving the 9 and -x to the other side becomes 5x² x - 9. D Since D 0 the equation will have two real roots and distinct roots.

The discriminant value which is part of the formula for solving the quadratic equation help us understand the nature of roots. B2 4ac 0. B² 4 a c 0 -7² -4 k2 49 8k 0.

So the equation with these roots is. A complex or imaginary number is denoted by i eg. So the equation with these roots is.

For the four quadratic equations given above the roots are as given below. Nature of Roots of the Quadratic Equation Example. If a_2.

Hence the other root of the required equation is 3 - 2i Since the complex roots always occur in pairs so other root. The quadratic equation will have imaginary roots ie α p iq and β p iq. 1 1 4i73 265 85 7 1 321 20 1 - 321 20 9 -1 i11 2 -1 - i11 2 11 2 - 7 3 13 2i33 11 - 2i33 11 15 0.

8k 49. Nature of the Roots of the Quadratic Equations. X 12 and x 2 Equation 4.

If b 2 4ac 0 roots are imaginary or you can say complex roots. Comparing the equation with the general form ax 2 bx c 0 gives a 1 b -5 and c 6. Where iq is the imaginary part of a complex number.

Nature of the Roots of a Quadratic Equation Examples. These complex roots will always occur in pairs ie both the roots are conjugate of each other. X 2 and x 3 Equation 2.

A quadratic equation in its standard form is represented as. 151 x 2 6 x 3 0 a 1 b 6 c 3 6 6 2 4 1 3 2 1 6 36 12 2 6 24 2 6 4899 2. Some methods for finding the roots are.

Assume that the quadratic polynomial has somewhere a negative value. So a quadratic equation has two roots. As example 8x 2 5x 10 0 is a quadratic equation.

D b 2 - 4ac. For example in using the quadratic formula to calculate the the roots of the equation x 2 6 x 3 0 the discriminant is positive and we will end up with two real-valued roots. On comparing with ax² bx c0.

Examples of quadratic equations in other forms include. Find the quadratic equation with real coefficients which has 3 - 2i as a root i -1. Which is a negative indicating that the roots of the quadratic equation are imaginary.

Use the quadratic formula to find the roots of x 2 -5x6 0. If the value of discriminant 0 ie. X sum of the roots x product of the roots 0 if and ᵦ be the two roots of a quadratic equation are given then the formula to form the quadratic equation is given by.

Ax2 bx c 0 where ab and c are real numbers such that a 0 and x is a variable. Since roots are equal. Since a quadratic polynomial has at most two zeros it cant have further imaginary roots.

All the quadratic equations. The quadratic equation will have equal roots ie. From given equation kx² 7x 20.

It is imaginary because the term under the square root is negative. X2 0x 9 0. X 1 -b b2-4ac2a.

P 3i3i 9. Is less than zero and the quadratic equation whose roots were finding is said to have complex or imaginary roots. αβ 3i.

X 2 5x 6. Answers to Quadratic Equations with Imaginary Roots ID. B 2 4ac 0 and perfect square.

Find the value of K such that the quadratic equation kx² 7x 2 0 has equal roots. D 5 2 4 x 1 x 6 25 -24 1. The given equation is of the form a x2 bx c 0.

The discriminant of a quadratic equation. That is we will analyse whether the roots of a quadratic equation are equal or unequal real or imaginary and rational or irrational. When b 2 4ac is negative the roots are complex imaginary.

B2 4ac 0. If we strictly answer the question and require imaginary roots then we have no real component so. In this case we say that the roots are imaginary.

Then the discriminant of the given equation is. X -2 and x 3 Equation 3. Two imaginary solutions 19 16.

Then by Mean Value Theorem you will find two real zeros. Below given the nature of the roots of the quadratic equation example will help you to understand the concept thoroughly. 5 12.

4i is an imaginary number 4. In this section we will examine the roots of a quadratic equation. P 2 i2 i 4 1 5.

A polynomial equation whose degree is 2 is known as quadratic equation. These roots may be real or imaginary. Let the quadratic equation be x 2 6x110.

X 1 and x 32 In general the roots of a quadratic equation. Let us understand the concept by solving some nature of roots of a quadratic equation practices problem. One real solution17 -180.

α β -b2a. According to the problem coefficients of the required quadratic equation are real and its one root is 3 - 2i. When a b and c are real numbers a 0 and the discriminant is negative then the roots α and β of the quadratic equation ax 2 bx c 0 are unequal and not real.

We get a k b -7 c 2. So Discriminant D 0. Solved example to find the imaginary roots occur in conjugate pairs of a quadratic equation.

A quadratic equation in its standard form is represented as. If the value of discriminant 0 ie. Given the quadratic equation look at only to find the roots.

Substitute the values in the quadratic formula. The type of number for the final answer rationalirrationalrealimaginary x2 - 6x 9 0 x2 - 4x 3 0 x2 - 4x - 3 0 x2 - 4x 7 0 Choose the appropriate words from the table to describe the roots obtained for the equations above. S 3i 3i 0.

The nature of the roots of a quadratic equation can be determined without actually finding the problems roots α β. The number of roots of a polynomial equation is equal to its degree. B 2 4ac -52 416 1.

Completing the square method.

How To Use The Quadratic Formula To Find Roots Of Equations Video Lesson Transcript Study Com