Finding the Quadratic Equation from Roots
To find the quadratic equation given its roots, we use the relationship between the roots and the coefficients of the equation. If the roots of a quadratic equation are α and β, the equation can be written as: $x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0$ or $x^2 - (\alpha + \beta)x + \alpha\beta = 0$
Identify Given Roots
The given roots are:
- Root 1: α = -2
- Root 2: β = 3
Calculate Sum and Product of Roots
Calculate the sum and product using the given roots:
- Sum of roots:
$\alpha + \beta = -2 + 3 = 1$ - Product of roots:
$\alpha \beta = (-2) \times 3 = -6$
Form the Quadratic Equation
Substitute the calculated sum and product into the standard equation format:
$x^2 - (\alpha + \beta)x + \alpha\beta = 0$
$x^2 - (1)x + (-6) = 0$
$x^2 - x - 6 = 0$
This equation matches option 2.
Similar Questions
- If the roots of the equation $(4 + m)x^2 + (m + 1)x + 1 = 0$ are equal, then find the values of m.
- The possible value of x in the equation $\sqrt{\frac{x}{1-x}} + \sqrt{\frac{1-x}{x}} = \frac{25}{12}$ is:
- The roots of the equation $y^2 - \sqrt{5} y - y + \sqrt{5} = 0$ are:
- If one of the roots of the equation $x^2 - 19x + 88 = 0$ is 8, then what is the other root?
- The roots of the equation $3x^{2} + 5x - a = 0$ are the reciprocals of each other. What is the value of $a$?
- Solve the following :
$\sqrt{7 + \sqrt{7 + \sqrt{7 + \sqrt{7 + \cdots}}}}$ The sum of two positive numbers is 14 and their product is 45. The positive difference between them is:
The sum of two positive numbers is 13 and their product is 30. The positive difference between them is:
The sum of two positive numbers is 35 and their product is 286. The positive difference between them is:
- Which of the quadratic equations below will not have real roots?
Important Questions from Quadratic Equation
For what values of k, the roots of 9x 2 + 8kx + 16 = 0 are real and equal?
- If \(\rm \left( \frac{x}{x+1} \right)^2 -5 \left( \frac{x}{x+1} \right) +6=0 \) , then the value of \(\rm \left( 1+\frac{1}{x} \right) \) is equal to :
The nature of the roots of the equation 4x 2 - 2x - 3 = 0.
If the roots of the equation (q – r)x 2+ (r – p)x + (p – q) = 0 are equal, then which of the following is true?
Number of real roots of the quadratic equation 3x 2+ 4x + 25 = 0 is
