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)?
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Solving Quadratic Equation Roots Problem
The problem asks us to find the value of \((p - q)\) given a quadratic equation \(px^2 + 3x + 2q = 0\) where the sum and product of its roots are both equal to -6.
Understanding Quadratic Equation Roots
For a standard quadratic equation in the form \(ax^2 + bx + c = 0\), the sum of the roots (\(\alpha + \beta\)) and the product of the roots (\(\alpha \beta\)) are related to the coefficients by the following formulas:
- Sum of roots: \(\alpha + \beta = -\frac{b}{a}\)
- Product of roots: \(\alpha \beta = \frac{c}{a}\)
Applying Formulas to the Given Equation
The given quadratic equation is \(px^2 + 3x + 2q = 0\). Comparing this to the standard form \(ax^2 + bx + c = 0\), we have:
- \(a = p\)
- \(b = 3\)
- \(c = 2q\)
According to the problem statement, the sum of the roots is -6, and the product of the roots is also -6.
Using the Sum of Roots Information
The sum of roots is \(-\frac{b}{a}\). So, we have:
\(\text{Sum of roots} = -\frac{3}{p}\)
We are given that the sum of roots is -6. Therefore:
\(-\frac{3}{p} = -6\)
To solve for \(p\), we can cross-multiply or rearrange the equation:
\(-3 = -6p\)
\(p = \frac{-3}{-6}\)
\(p = \frac{1}{2}\)
So, the value of \(p\) is \(\frac{1}{2}\).
Using the Product of Roots Information
The product of roots is \(\frac{c}{a}\). So, we have:
\(\text{Product of roots} = \frac{2q}{p}\)
We are given that the product of roots is -6. Therefore:
\(\frac{2q}{p} = -6\)
Now, substitute the value of \(p = \frac{1}{2}\) that we found:
\(\frac{2q}{1/2} = -6\)
Simplifying the left side:
\(2q \times 2 = -6\)
\(4q = -6\)
To solve for \(q\):
\(q = \frac{-6}{4}\)
\(q = -\frac{3}{2}\)
So, the value of \(q\) is \(-\frac{3}{2}\).
Calculating (p - q)
The question asks for the value of \((p - q)\). We have \(p = \frac{1}{2}\) and \(q = -\frac{3}{2}\).
\(p - q = \frac{1}{2} - \left(-\frac{3}{2}\right)\)
\(p - q = \frac{1}{2} + \frac{3}{2}\)
\(p - q = \frac{1 + 3}{2}\)
\(p - q = \frac{4}{2}\)
\(p - q = 2\)
Thus, the value of \((p - q)\) is 2.
| Step | Calculation | Result |
|---|---|---|
| 1 | Set sum of roots equal to -6: \(-\frac{3}{p} = -6\) | \(p = \frac{1}{2}\) |
| 2 | Set product of roots equal to -6: \(\frac{2q}{p} = -6\) | \(\frac{2q}{1/2} = -6 \implies 4q = -6\) |
| 3 | Solve for q | \(q = -\frac{6}{4} = -\frac{3}{2}\) |
| 4 | Calculate \((p - q)\) | \(\frac{1}{2} - \left(-\frac{3}{2}\right) = \frac{1}{2} + \frac{3}{2} = \frac{4}{2} = 2\) |
Revision Table: Key Concepts for Quadratic Equations
| Concept | Formula for \(ax^2 + bx + c = 0\) |
|---|---|
| Sum of Roots (\(\alpha + \beta\)) | \(-\frac{b}{a}\) |
| Product of Roots (\(\alpha \beta\)) | \(\frac{c}{a}\) |
| Discriminant (\(\Delta\) or \(D\)) | \(b^2 - 4ac\) |
| Roots of the equation (Quadratic Formula) | \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) |
Additional Information: Properties of Quadratic Roots
The sum and product of roots are powerful tools for analyzing quadratic equations without actually finding the roots themselves. They are derived directly from the quadratic formula. Understanding these relationships helps in solving problems involving roots' properties, forming quadratic equations when roots are given, and determining the nature of roots based on the discriminant.
- If \(\alpha\) and \(\beta\) are the roots of \(ax^2 + bx + c = 0\), then the equation can also be written as \(x^2 - (\alpha + \beta)x + \alpha \beta = 0\).
- This means \(a(x^2 - (\alpha + \beta)x + \alpha \beta) = 0\), which expands to \(ax^2 - a(\alpha + \beta)x + a(\alpha \beta) = 0\).
- Comparing coefficients with \(ax^2 + bx + c = 0\), we get \(b = -a(\alpha + \beta)\) (so \(\alpha + \beta = -b/a\)) and \(c = a(\alpha \beta)\) (so \(\alpha \beta = c/a\)).
- These relationships hold true regardless of whether the roots are real or complex.
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