Finding the Quadratic Equation from Sum and Product of Roots
The problem asks us to find a quadratic equation given the sum and product of its roots.
Standard Form of a Quadratic Equation
A quadratic equation can be generally represented in the form:
$$x^2 - (\text{Sum of Roots})x + (\text{Product of Roots}) = 0$$
Let the sum of the roots be denoted by $S$ and the product of the roots be denoted by $P$. The equation is then:
$$x^2 - Sx + P = 0$$
Given Values
From the question, we have:
- Sum of the roots, $S = (4-3\sqrt{2})$
- Product of the roots, $P = -28$
Constructing the Equation
Now, we substitute these values into the standard form:
$$x^2 - (4-3\sqrt{2})x + (-28) = 0$$
Simplifying the equation, we get:
$$x^2 - (4-3\sqrt{2})x - 28 = 0$$
Matching with Options
Comparing this result with the given options:
- Option 1: $x^2-(4+3\sqrt{2})x+28=0$
- Option 2: $x^2+(4+3\sqrt{2})x+28=0$
- Option 3: $x^2+(4-3\sqrt{2})x-28=0$
- Option 4: $x^2-(4-3\sqrt{2})x-28=0$
Our derived equation matches Option 4.
Important Questions from Quadratic equation
What number should be subtracted from x3−4x2−8x+11 to make the number divisible by (x+2)?
- Find the value of K if the quadratic equations $2x^2 + Kx + 8 = 0$ and $3x^2 + 4x + 12 = 0$ have both roots common.
- If the quadratic equations $4x^2 + bx + 3 = 0$ and $8x^2 + 4x + c = 0$ have both the roots common, find the values for b and c, respectively.
- Determine the nature of the roots of the quadratic equation $3x^2 + 2x + 5 = 0$.
- Find roots of $5m^2 + 18m + 16 = 0$
