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 ?
x 2 - 8x + 12 = 0
Understanding the Quadratic Equation and its Roots
A quadratic equation is a polynomial equation of the second degree. The standard form is \(ax^2 + bx + c = 0\), where \(a, b, \text{ and } c\) are coefficients and \(a \neq 0\). The roots of a quadratic equation are the values of \(x\) that satisfy the equation.
For a quadratic equation \(ax^2 + bx + c = 0\) with roots \(\alpha\) and \(\beta\), the following relationships hold:
- Sum of roots: \(\alpha + \beta = -\frac{b}{a}\)
- Product of roots: \(\alpha \beta = \frac{c}{a}\)
Conversely, if you know the sum and product of the roots, say \(S\) and \(P\), the quadratic equation can be written as \(x^2 - Sx + P = 0\).
Finding the Original Quadratic Equation and its Roots
We are given that the sum of the roots of the original quadratic equation is 7, and the product of the roots is 12.
- Sum of original roots \(S_{original} = 7\)
- Product of original roots \(P_{original} = 12\)
The original quadratic equation is therefore:
\(x^2 - S_{original}x + P_{original} = 0\)
\(x^2 - 7x + 12 = 0\)
To find the original roots, we can solve this equation. We look for two numbers that add up to 7 and multiply to 12. These numbers are 3 and 4.
So, the equation can be factored as:
\((x - 3)(x - 4) = 0\)
Setting each factor to zero gives the roots:
- \(x - 3 = 0 \implies x = 3\)
- \(x - 4 = 0 \implies x = 4\)
The original roots are 3 and 4. The bigger root is 4, and the smaller root is 3.
Transforming the Roots
According to the question, the original roots are transformed:
- The bigger root is halved. Original bigger root is 4. New bigger root is \(4 / 2 = 2\).
- The smaller root is doubled. Original smaller root is 3. New smaller root is \(3 \times 2 = 6\).
The resulting roots are 2 and 6.
Determining the Resulting Quadratic Equation
Now we need to find the quadratic equation whose roots are 2 and 6. Let the new roots be \(\alpha'\) and \(\beta'\). So, \(\alpha' = 2\) and \(\beta' = 6\).
First, find the sum and product of these new roots:
- Sum of new roots \(S_{new} = 2 + 6 = 8\)
- Product of new roots \(P_{new} = 2 \times 6 = 12\)
The resulting quadratic equation with roots \(\alpha'\) and \(\beta'\) is given by:
\(x^2 - S_{new}x + P_{new} = 0\)
Substitute the sum and product of the new roots:
\(x^2 - 8x + 12 = 0\)
This is the resulting quadratic equation.
Summary of Steps
Here is a brief summary of the process:
- Use the given sum and product to find the original quadratic equation.
- Solve the original quadratic equation to find its roots.
- Identify the bigger and smaller original roots.
- Apply the given transformations (halving the bigger root, doubling the smaller root) to find the new roots.
- Use the sum and product of the new roots to form the resulting quadratic equation.
Revision Table: Key Concepts
| Concept | Description | Formula/Example |
|---|---|---|
| Quadratic Equation | An equation of the form \(ax^2 + bx + c = 0\), \(a \neq 0\) | \(x^2 - 7x + 12 = 0\) |
| Roots of an Equation | Values of the variable that satisfy the equation | For \(x^2 - 7x + 12 = 0\), roots are 3 and 4 |
| Sum of Roots | For \(ax^2 + bx + c = 0\), sum is \(-b/a\) | For \(x^2 - 7x + 12 = 0\), sum is \(-(-7)/1 = 7\) |
| Product of Roots | For \(ax^2 + bx + c = 0\), product is \(c/a\) | For \(x^2 - 7x + 12 = 0\), product is \(12/1 = 12\) |
| Forming Eq. from Roots | If roots are \(\alpha, \beta\), equation is \(x^2 - (\alpha+\beta)x + \alpha\beta = 0\) | If roots are 2, 6, eq. is \(x^2 - (2+6)x + (2 \times 6) = x^2 - 8x + 12 = 0\) |
Additional Information: Properties of Quadratic Roots
The relationship between the coefficients of a quadratic equation and its roots is a fundamental concept. This allows us to determine properties of the roots without directly solving the equation, or to construct the equation if the roots (or their sum/product) are known.
For the standard form \(ax^2 + bx + c = 0\), the discriminant, given by \(\Delta = b^2 - 4ac\), tells us about the nature of the roots:
- If \(\Delta > 0\), the roots are real and distinct.
- If \(\Delta = 0\), the roots are real and equal.
- If \(\Delta < 0\), the roots are complex (non-real) conjugates.
In our case, for \(x^2 - 7x + 12 = 0\), \(a=1, b=-7, c=12\). \(\Delta = (-7)^2 - 4(1)(12) = 49 - 48 = 1\). Since \(\Delta > 0\), the roots are real and distinct (which we found to be 3 and 4). For the resulting equation \(x^2 - 8x + 12 = 0\), \(a=1, b=-8, c=12\). \(\Delta = (-8)^2 - 4(1)(12) = 64 - 48 = 16\). Since \(\Delta > 0\), the roots are real and distinct (which we found to be 2 and 6).
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Important Questions from Quadratic Equation
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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 ?
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