Question

Find the nature of roots of the quadratic equation, 2x2-11x + 9 = 0

Answer 1

Given:

2x²-11x+9=0

To Find:

Nature of roots of the given quadratic equation.

Solution:

2x²-11x+9=0......(Given)

Comparing the above equation with

ax²+bx+c

So, a=2 , b= -11 and c=9

b²-4ac(∆) =(-11) ²-4×2×9

∆ = 121-72

∆ =49

Here, ∆>0

So the roots of the given quadratic equation are real and unequal.

For More Information:

1) ∆<0→ Roots are not real.

2) ∆=0→ Roots are real and equal.

3) ∆>0→ Roots are real and unequal.

Answer 2

Answer:

The nature of roots of the given quadratic equation is real and unequal.

Step-by-step-explanation:

The given quadratic equation is

2x² - 11x + 9 = 0.

Comparing with ax² + bx + c = 0, we get,

• a = 2

• b = - 11

• c = 9

Δ = b² - 4ac

➞ ( - 11 )² - 4 × 2 × 9

➞ 121 - 8 × 9

➞ 121 - 72

➞ Δ = b² - 4ac = 49

Here,

Δ = 49 > 0

∴ The nature of roots of the given quadratic equation is real and unequal.

Additional Information:

1. Quadratic Equation :

An equation having a degree '2' is called quadratic equation.

The general form of quadratic equation is

ax² + bx + c = 0

Where, a, b, c are real numbers and a ≠ 0.

2. Roots of Quadratic Equation:

The roots means nothing but the value of the variable given in the equation.

3. Methods of solving quadratic equation:

There are mainly three methods to solve or find the roots of the quadratic equation.

A) Factorization method

B) Completing square method

C) Formula method

4. Formula to solve quadratic equation:

$\boxed{\red{\sf\:x\:=\:\dfrac{-\:b\:\pm\:\sqrt{b^{2}\:-\:4ac}}{2a}}}$

5. Determining the nature of roots of a quadratic equation:

1. The nature of the roots of the quadratic equation

ax² + bx + c = 0 depends on the value of b² - 4ac.

2. b² - 4ac is called the determinant. It is denoted by $\sf\:\triangle$ ( The Greek letter Delta ).

3. Value of Δ and nature of roots:

$\begin{array}{|c|c|}\cline{1-2}{\triangle = 0 & Real\:and\:equal}\\\cline{1-2}{\triangle > 0 & Real\:and\:unequal}\\\cline{1-2}{\triangle < 0 & Not\:real}\\\cline{1-2}\end{array}$