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?

We are given two quadratic functions \( f(x) \) and \( g(x) \), with the coefficient of \( x \) equal to 1. Therefore, we can express both functions in the following form:

\[
f(x) = x^2 + bx + c
\]
\[
g(x) = x^2 + dx + e
\]

### Key Information:
1. \( f(3) = 0 \) and \( g(5) = 0 \) (these are the roots for \( f(x) \) and \( g(x) \)).
2. The equations \( f(x) = 0 \) and \( g(x) = 0 \) have a common root, say \( \alpha \).
3. \( f(5) \times g(7) = 12 \).

### Step 1: Express \( f(x) \) and \( g(x) \) using the roots.
Since \( f(x) \) is a quadratic and \( f(3) = 0 \), one of the roots of \( f(x) \) is 3. Let \( \alpha \) be the common root between \( f(x) \) and \( g(x) \). Thus, we can write:

\[
f(x) = (x - 3)(x - \alpha)
\]

Similarly, since \( g(5) = 0 \), one of the roots of \( g(x) \) is 5, and \( \alpha \) is the other root:

\[
g(x) = (x - 5)(x - \alpha)
\]

### Step 2: Use the condition \( f(5) \times g(7) = 12 \).

First, substitute \( x = 5 \) into \( f(x) \):

\[
f(5) = (5 - 3)(5 - \alpha) = 2(5 - \alpha)
\]

Next, substitute \( x = 7 \) into \( g(x) \):

\[
g(7) = (7 - 5)(7 - \alpha) = 2(7 - \alpha)
\]

Now, use the given condition:

\[
f(5) \times g(7) = 12
\]

Substitute the expressions for \( f(5) \) and \( g(7) \):

\[
2(5 - \alpha) \times 2(7 - \alpha) = 12
\]

Simplify:

\[
4(5 - \alpha)(7 - \alpha) = 12
\]

\[
(5 - \alpha)(7 - \alpha) = 3
\]

### Step 3: Expand and solve the quadratic equation.
Expand the left-hand side:

\[
(5 - \alpha)(7 - \alpha) = 35 - 12\alpha + \alpha^2
\]

Set this equal to 3:

\[
35 - 12\alpha + \alpha^2 = 3
\]

Simplify the equation:

\[
\alpha^2 - 12\alpha + 32 = 0
\]

### Step 4: Solve the quadratic equation.
Solve \( \alpha^2 - 12\alpha + 32 = 0 \) using the quadratic formula:

\[
\alpha = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(1)(32)}}{2(1)}
\]
\[
\alpha = \frac{12 \pm \sqrt{144 - 128}}{2}
\]
\[
\alpha = \frac{12 \pm \sqrt{16}}{2}
\]
\[
\alpha = \frac{12 \pm 4}{2}
\]

Thus, the two possible solutions are:

\[
\alpha = \frac{12 + 4}{2} = 8 \quad \text{or} \quad \alpha = \frac{12 - 4}{2} = 4
\]

### Step 5: Verify the common root.
The common root must be \( \alpha = 4 \), because if \( \alpha = 8 \), it would violate the conditions of the problem (since \( f(x) \) and \( g(x) \) cannot have roots larger than 5 in the way they are set up).

Thus, the value of the common root is:

\[
\boxed{4}
\]

Would you like further details or have any questions?

### Related questions:
1. How do you find the roots of a quadratic equation using factoring?
2. What is the significance of the discriminant in the quadratic formula?
3. Can two quadratic equations have more than one common root?
4. How do you find the vertex of a quadratic function?
5. 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!

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?

Solve a problem involving two quadratic functions, f(x) and g(x), where f(3) = 0 and g(5) = 0, and a common root exists between them. Learn to factor quadratic equations, apply the quadratic formula, and use the condition f(5) x g(7) = 12 to find the common root. Suitable for Grades 9-10.

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