If α and β are the roots of the quadratic equation x 2+ kx – 15 = 0 such that α – β = 8, then what is the positive value of k?
2
Solving Quadratic Equation Roots Problem
We are given a quadratic equation and information about its roots. The goal is to find the positive value of the coefficient 'k'.
The given quadratic equation is: \(x^2 + kx - 15 = 0\)
Let the roots of this quadratic equation be \(\alpha\) and \(\beta\).
Understanding Roots of a Quadratic Equation
For a standard quadratic equation \(ax^2 + bx + c = 0\), the relationships between the roots (\(\alpha\) and \(\beta\)) and the coefficients (a, b, c) are:
- Sum of roots: \(\alpha + \beta = -\frac{b}{a}\)
- Product of roots: \(\alpha \beta = \frac{c}{a}\)
Applying Root Relationships to the Given Equation
In our equation \(x^2 + kx - 15 = 0\), we have:
- \(a = 1\)
- \(b = k\)
- \(c = -15\)
Using the relationships above:
- Sum of roots: \(\alpha + \beta = -\frac{k}{1} = -k\)
- Product of roots: \(\alpha \beta = \frac{-15}{1} = -15\)
We are also given an additional condition:
- Difference of roots: \(\alpha - \beta = 8\)
Calculating the Value of k
We have three pieces of information:
- \(\alpha + \beta = -k\)
- \(\alpha \beta = -15\)
- \(\alpha - \beta = 8\)
We can use an algebraic identity that relates the sum, difference, and product of two numbers. The identity is:
\((\alpha + \beta)^2 = (\alpha - \beta)^2 + 4\alpha \beta\)
Now, substitute the values we know into this identity:
\((-k)^2 = (8)^2 + 4(-15)\)
Simplify the equation:
\(k^2 = 64 - 60\)
\(k^2 = 4\)
To find the value of k, take the square root of both sides:
\(k = \pm\sqrt{4}\)
\(k = \pm 2\)
So, the possible values for k are 2 and -2.
Finding the Positive Value of k
The question asks for the positive value of k. Between the two possible values, 2 and -2, the positive value is 2.
Therefore, the positive value of k is 2.
Summary of Calculation Steps
| Step | Description | Calculation |
|---|---|---|
| 1 | Identify coefficients a, b, c | \(a=1, b=k, c=-15\) |
| 2 | Write sum of roots in terms of k | \(\alpha + \beta = -k\) |
| 3 | Write product of roots | \(\alpha \beta = -15\) |
| 4 | Use the given difference of roots | \(\alpha - \beta = 8\) |
| 5 | Apply identity \((\alpha + \beta)^2 = (\alpha - \beta)^2 + 4\alpha \beta\) | \((-k)^2 = (8)^2 + 4(-15)\) |
| 6 | Solve for k | \(k^2 = 64 - 60 = 4 \implies k = \pm 2\) |
| 7 | Select the positive value of k | \(k = 2\) |
The positive value of k that satisfies the given conditions for the quadratic equation \(x^2 + kx - 15 = 0\) with roots \(\alpha\) and \(\beta\) such that \(\alpha - \beta = 8\) is 2.
Revision Table: Quadratic Equation Roots
| Concept | Formula/Relationship | Notes |
|---|---|---|
| Standard Form | \(ax^2 + bx + c = 0\) | a, b, c are coefficients, \(a \ne 0\) |
| Sum of Roots (\(\alpha + \beta\)) | \(-\frac{b}{a}\) | Relationship between roots and coefficients |
| Product of Roots (\(\alpha \beta\)) | \(\frac{c}{a}\) | Relationship between roots and coefficients |
| Relationship between Sum, Difference, Product | \((\alpha + \beta)^2 = (\alpha - \beta)^2 + 4\alpha \beta\) | Useful identity for problems like this |
Additional Information: Solving Quadratic Equations
Solving quadratic equations involves finding the values of x that satisfy the equation. These values are called the roots or zeros of the equation. There are several methods to find the roots:
- Factoring: If the quadratic expression can be factored, set each factor to zero and solve for x.
- Completing the Square: A method to transform the equation into the form \((x-h)^2 = p\), which can then be solved by taking the square root.
- Quadratic Formula: This formula gives the roots directly from the coefficients a, b, and c: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). The expression under the square root, \(b^2 - 4ac\), is called the discriminant, which tells us about the nature of the roots (real, imaginary, distinct, repeated).
The problem discussed here uses the properties of the roots (sum and product) rather than finding the roots directly, which is often useful when dealing with relationships between roots and coefficients.
Similar Questions
For what values of k, the roots of 9x 2 + 8kx + 16 = 0 are real and equal?
If the sum of the squares of the roots of the equation x 2- 14x + k = 0 is 100, then what is the value of k ?
Consider a question and two statements:
Question :
Does the equation ax 2+ bx + c = 0 have real roots of opposite sign?
Statement – I : The discriminant D > 0
Statement – II : c / a > 0
Which one of the following is correct in respect of the question and the statements?
What is the value of α (α ≠ 0) for which x 2– 5x + α and x 2– 7x + 2α have a common factor?
Let α and β be the roots of the equation \(\rm \frac{1}{x+a+b}=\frac{1}{x}+\frac{1}{a}+\frac{1}{b}\); a ≠ 0, b ≠ 0, x ≠ 0.
Which one of the following is a quadratic equation whose roots are α2 and β2?
Which one of the following equations does not have real roots ?
If p and q (p > q) are the roots of the equation x 2 - 60x + 899 = 0, then which one of the following is correct ?
If \(\frac{x}{a} + \frac{y}{b} = a + b\) and \(\frac{x}{a^2} + \frac{y}{b^2} = 2\) , then what is \(\frac{x}{a^2} - \frac{y}{b^2}\) equal to?
The sum and the product of the roots of a quadratic equation are 7 and 12 respectively. If the bigger root is halved and the smaller root is doubled, then what is the resulting quadratic equation ?
Two numbers p and q are such that the quadratic equation px 2+ 3x + 2q = 0 has – 6 as the sum and the product of the roots. What is the value of (p – q)?
Important Questions from Quadratic Equation
For what values of k, the roots of 9x 2 + 8kx + 16 = 0 are real and equal?
If the sum of the squares of the roots of the equation x 2- 14x + k = 0 is 100, then what is the value of k ?
- 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?
