find the nature of the roots of the following quadratic equation if real roots are exist then find them 2x2-3x+1=0
Answers
Answer:
Given :-
- Quadratic equation if real roots are exist is 2x² - 3x + 1 = 0.
To Find :-
- What is the nature of the roots.
Formula Used :-
➲ Discriminate (D) = b² - 4ac
Solution :-
Given equation :
➦ 2x² - 3x + 1 = 0
where,
◈ a = 2
◈ b = - 3
◈ c = 1
Then,
↦ Discriminate (D) = b² - 4ac
↦ Discriminate (D) = (- 3)² - 4(2)(1)
↦ Discriminate (D) = (- 3)(- 3) - 8(1)
↦ Discriminate (D) = 9 - 8
➠ Discriminate (D) = 1 > 0
∴ The nature of the two distinct real roots.
__________________________
Extra Information :
★ The general form of equation is a x² + bx + c.
[ If a = 0, then the equation becomes to a linear equation.
If b = 0, then the roots of the equation becomes equal but opposite in sign.
If c = 0, then one of the roots is zero. ]
★ b² - 4ac is the discriminate of the equation.
◆ When b² - 4ac = 0 then roots are real and equal.
◆ When, b² - 4ac > 0 then the roots are imaginary and unequal.
◆ When, b² - 4ac < 0 then there will be no real roots.
Basic concepts:
- D > 0, two distinct real roots.
- D = 0, two equal roots.
- D < 0, no real roots.
Solution:
Given that,
We are given with a quadratic equation 2x² - 3x + 1 = 0, and we need to find out the nature of the roots.
So,
The given equation is 2x² - 3x + 1 = 0.
Comparing this equation with ax² + bx +c = 0, we get :
- a = 2
- b = -3
- c = 1.
Now, we know that, if we are given with a, b and c, then we have the required formula, that is,
→ D = b²- 4ac
Substituting all the given values in the formula, we get:
→ D = (-3)² - 4 * 2 * 1
→ D = 9 - 4 * 2 * 1
→ D = 9 - 8 * 1
→ D = 9 - 8
→ D = 1
Since, D > 1.
Hence, there are two distinct real roots.
Extra information:
Quadratic equation =>
A quadratic equation in the variable x is an equation form ax² - bx + x = 0, where a, b, c are real numbers and a ≠ 0.
Example =>
This is the example of quadratic equation,
9x² + 7x - 2 = 0.