In order to find the sign of roots we use sign given in the equation.
For example: x^{2} + 3x – 4 = 0
The factors are 4 and 1
Now will find the sign of roots. According to the table, if sign given in the equation is + and – then their sign of roots is – and +. Therefore, the roots of the equation are 4 and 1
When one equation has positive roots while the other has negative roots, or when both equations have roots of the same sign, compare the roots to determine the relationship between them.
Q1: Find the correct option given below by solving the following equation x^{2} – 7x + 10 = 0 and y^{2} + 8y + 15 = 0
(a) x < y
(b) x > y
(c) x ≤ y
(d) x ≥ y
Ans: (b)
x^{2} – 7x + 10 = 0…. (1)
Roots of first equation are 5 and 2
Using the table above, we will find the sign of roots. If sign given in the equation is – and + then their sign of roots is + and +
Therefore, the roots of the equation are 5 and 2
Now, y^{2} + 8y + 15 = 0 ……. (2)
Roots of second equation are 3 and 5
If sign given in the equation is + and + then their sign of roots is – and –
Therefore, the roots of the equation are 3 and – 2
Now, here roots are +x_{1}, + x_{2}, – y_{1}, and – y_{2}
This clearly shows that X roots are positive and Y roots are negative. So, x> y
When roots of the equation are positive and negative and we cannot find the relation between them.
Q2: Find the correct option given below by solving the following equation 6x^{2} + 11x – 35 = 0 and 6y^{2} + 5y – 6 = 0
(a) x < y
(b) x > y
(c) x ≤ y
(d) cannot be determined
Ans: (d)
6x^{2} + 11x – 35 = 0…. (1)
Roots of first equation are 3.5 and 1.66
Using the table above, we will find the sign of roots. If sign given in the equation is + and – then their sign of roots is – and +
Therefore, the roots of the equation are – 3.5 and 1.66
Now, 6y^{2} + 5y – 6 = 0 ……. (2)
Roots of second equation are 1.5 and 0.66
If sign given in the equation is + and – then their sign of roots is – and +
Therefore, the roots of the equation are – 1.5 and 0.66
Now, here roots are x_{1}, + x_{2}, – y_{1}, and + y_{2}
It means x_{2}> – y_{1} and –x_{1}< + y_{2}
So, we cannot determine which roots are greater.
When we can find a equation where x is a variable and a, b, and c represent constants and D, i.e., Discriminant.
The discriminant is the part of the quadratic formula underneath the square root symbol: b²4ac.
Q3: Find the Discriminant and roots of the following equation: 3x2 − 5x − 7 = 0.
Sol:
Let us find Discriminant,
D = 109
So the two roots are 2.57 and 0.90
Q4: Which method is used to solve quadratic equations that cannot be easily factored or involve complex coefficients?
(a) Factoring
(b) Quadratic Formula
(c) Completing the Square
(d) Estimation
Ans: (b)
The quadratic formula is a universal method used to solve any quadratic equation, regardless of its coefficients or factorability. It provides precise solutions by directly calculating the roots.
Q5: For the quadratic equation 3x² + 6x – 9 = 0, what are the roots?
(a) x = 1
(b) x = 1
(c) x = 3
(d) x = 3
Ans: (c)
Divide the entire equation by 3 to simplify it: x² + 2x – 3 = 0. This equation can be factored as (x – 1)(x + 3) = 0. Setting each factor to zero gives x – 1 = 0 and x + 3 = 0. Solving for “x” in both equations yields x = 1 and x = 3. Therefore, the roots are x = 3 and x = 1.
314 videos170 docs185 tests

1. What is a quadratic equation? 
2. How do you solve a quadratic equation using the quadratic formula? 
3. Are there any shortcuts to solve quadratic equations? 
4. Can all quadratic equations be solved using shortcuts? 
5. What are some tips for solving quadratic equations efficiently? 
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