If one root of a quadratic equation is `-2`. At `x= -5,y=27` and x = 7, y = 27. Then what's the minimum value of the equation?

Options

A. 5
B. 0
C. `-9`
D. none of these

Answer

To solve the problem step by step, we start with the information given:

1. **Identify the Quadratic Equation**:
We know that one root of the quadratic equation is \( -2 \). Thus, we can express the quadratic equation in the form:
\[ y = a(x + 2)(x - r) \]
where \( r \) is the other root.

2. **Use Given Points**:
We are given that \( y = 27 \) when \( x = -5 \) and \( x = 7 \). We can use these points to form equations.

3. **Substituting the First Point**:
Substitute \( x = -5 \) into the quadratic equation:
\[ 27 = a(-5 + 2)(-5 - r) \]
Simplifying this gives:
\[ 27 = a(-3)(-5 - r) \quad \Rightarrow \quad 27 = 3a(5 + r) \quad \Rightarrow \quad 9 = a(5 + r) \quad \text{(Equation 1)} \]

4. **Substituting the Second Point**:
Now substitute \( x = 7 \):
\[ 27 = a(7 + 2)(7 - r) \]
Simplifying this gives:
\[ 27 = a(9)(7 - r) \quad \Rightarrow \quad 27 = 9a(7 - r) \quad \Rightarrow \quad 3 = a(7 - r) \quad \text{(Equation 2)} \]

5. **Solving the Two Equations**:
We now have two equations:
- \( 9 = a(5 + r) \) (Equation 1)
- \( 3 = a(7 - r) \) (Equation 2)

From Equation 1, we can express \( a \):
\[ a = \frac{9}{5 + r} \]
Substitute this value of \( a \) into Equation 2:
\[ 3 = \frac{9}{5 + r}(7 - r) \]
Cross-multiplying gives:
\[ 3(5 + r) = 9(7 - r) \]
Expanding both sides:
\[ 15 + 3r = 63 - 9r \]
Combining like terms:
\[ 12r = 48 \quad \Rightarrow \quad r = 4 \]

6. **Finding the Value of \( a \)**:
Substitute \( r = 4 \) back into Equation 1:
\[ 9 = a(5 + 4) \quad \Rightarrow \quad 9 = 9a \quad \Rightarrow \quad a = 1 \]

7. **Forming the Final Equation**:
Now we can write the quadratic equation:
\[ y = (x + 2)(x - 4) = x^2 - 2x - 8 \]

8. **Finding the Minimum Value**:
The vertex of a quadratic equation \( y = ax^2 + bx + c \) occurs at \( x = -\frac{b}{2a} \). Here, \( a = 1 \) and \( b = -2 \):
\[ x = -\frac{-2}{2 \cdot 1} = 1 \]
Substitute \( x = 1 \) into the equation to find the minimum value:
\[ y = 1^2 - 2(1) - 8 = 1 - 2 - 8 = -9 \]

Thus, the minimum value of the quadratic equation is \( -9 \).