Check that if the roots of the equation `(a^2+b^2)x^2+2x(ac+bd)+c^2+d^2=0`are real,whether they will be equal

Options

A. real and distinct
B. ral and equal
C. imaginary
D. nothing can be said

Answer

To determine if the roots of the quadratic equation \((a^2 + b^2)x^2 + 2x(ac + bd) + (c^2 + d^2) = 0\) are real and whether they are equal, we will use the discriminant method. The discriminant \(D\) of a quadratic equation \(Ax^2 + Bx + C = 0\) is given by:

\[ D = B^2 - 4AC \]

### Step 1: Identify the coefficients
In our equation, we can identify:
- \(A = a^2 + b^2\)
- \(B = 2(ac + bd)\)
- \(C = c^2 + d^2\)

### Step 2: Calculate the discriminant
Now, we will calculate the discriminant \(D\):

\[ D = B^2 - 4AC \]

Substituting the values of \(A\), \(B\), and \(C\):

\[ D = [2(ac + bd)]^2 - 4(a^2 + b^2)(c^2 + d^2) \]

### Step 3: Simplify the discriminant
Calculating \(B^2\):

\[ B^2 = 4(ac + bd)^2 \]

Now, substituting this back into the discriminant:

\[ D = 4(ac + bd)^2 - 4(a^2 + b^2)(c^2 + d^2) \]

Factoring out the common term \(4\):

\[ D = 4\left[(ac + bd)^2 - (a^2 + b^2)(c^2 + d^2)\right] \]

### Step 4: Analyze the expression
Next, we need to analyze the expression \((ac + bd)^2 - (a^2 + b^2)(c^2 + d^2)\).

Using the Cauchy-Schwarz inequality, we know:

\[ (ac + bd)^2 \leq (a^2 + b^2)(c^2 + d^2) \]

This means:

\[ (ac + bd)^2 - (a^2 + b^2)(c^2 + d^2) \leq 0 \]

### Step 5: Conclusion about the discriminant
Thus, we have:

\[ D \leq 0 \]

This implies that the roots of the quadratic equation are either real and equal (if \(D = 0\)) or complex (if \(D < 0\)).

### Step 6: Condition for equal roots
For the roots to be equal, we need \(D = 0\):

\[ (ac + bd)^2 = (a^2 + b^2)(c^2 + d^2) \]

### Final Conclusion
Therefore, if the roots of the equation are real, they will be equal if and only if:

\[ (ac + bd)^2 = (a^2 + b^2)(c^2 + d^2) \]