Math Problem Statement
f(x) and g(x) are quadratic functions with coefficient of x equal to 1, such that f(3) = g(5) = 0. If f(x) = 0 and g(x) = 0 have a common root and f(5) x g(7) = 12, what is the value of the common root?
Solution
We are given two quadratic functions and , with the coefficient of equal to 1. Therefore, we can express both functions in the following form:
Key Information:
- and (these are the roots for and ).
- The equations and have a common root, say .
- .
Step 1: Express and using the roots.
Since is a quadratic and , one of the roots of is 3. Let be the common root between and . Thus, we can write:
Similarly, since , one of the roots of is 5, and is the other root:
Step 2: Use the condition .
First, substitute into :
Next, substitute into :
Now, use the given condition:
Substitute the expressions for and :
Simplify:
Step 3: Expand and solve the quadratic equation.
Expand the left-hand side:
Set this equal to 3:
Simplify the equation:
Step 4: Solve the quadratic equation.
Solve using the quadratic formula:
Thus, the two possible solutions are:
Step 5: Verify the common root.
The common root must be , because if , it would violate the conditions of the problem (since and cannot have roots larger than 5 in the way they are set up).
Thus, the value of the common root is:
Would you like further details or have any questions?
Related questions:
- How do you find the roots of a quadratic equation using factoring?
- What is the significance of the discriminant in the quadratic formula?
- Can two quadratic equations have more than one common root?
- How do you find the vertex of a quadratic function?
- How do you determine if a quadratic function has real, repeated, or complex roots?
Tip:
When solving a quadratic equation, always check if factoring is possible before using the quadratic formula. It can save time!
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Roots of Quadratic Functions
Formulas
Standard form of a quadratic equation: f(x) = ax^2 + bx + c
Quadratic formula: x = [-b ± sqrt(b^2 - 4ac)] / 2a
Factored form of a quadratic function: f(x) = (x - root1)(x - root2)
Theorems
Quadratic Formula
Factoring of Quadratic Equations
Suitable Grade Level
Grades 9-10