Understanding the Quadratic Equation Problem

We are given a quadratic equation in the variable \(x\): \((a^2 + b^2) x^2 + 2(ac + bd) x + c^2 + d^2 = 0\). We need to determine the condition under which this quadratic equation has no real roots.

A standard form for a quadratic equation is \(Ax^2 + Bx + C = 0\), where \(A\), \(B\), and \(C\) are coefficients.

Identifying Coefficients

By comparing the given equation with the standard form \(Ax^2 + Bx + C = 0\), we can identify the coefficients:

• \(A = a^2 + b^2\)
• \(B = 2(ac + bd)\)
• \(C = c^2 + d^2\)

These coefficients are essential for applying concepts in mathematics related to quadratic equations.

Condition for No Real Roots

A quadratic equation has no real roots if and only if its discriminant is less than zero. The discriminant, denoted by \(D\), is calculated using the formula \(D = B^2 - 4AC\).

So, for no real solutions, we must satisfy \(D < 0\).

Calculating and Simplifying the Discriminant

Let's substitute the expressions for \(A\), \(B\), and \(C\) into the discriminant formula \(D = B^2 - 4AC\).

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

Now, we will simplify this expression step-by-step:

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

Divide the entire inequality by 4 (since 4 is positive, the inequality sign doesn't change):

\(\frac{D}{4} = (ac + bd)^2 - (a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2)\)

Expand the term \((ac + bd)^2\):

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

Substitute this back into the expression for \(\frac{D}{4}\):

\(\frac{D}{4} = (a^2c^2 + 2abcd + b^2d^2) - (a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2)\)

Distribute the negative sign:

\(\frac{D}{4} = a^2c^2 + 2abcd + b^2d^2 - a^2c^2 - a^2d^2 - b^2c^2 - b^2d^2\)

Cancel out the \(a^2c^2\) and \(b^2d^2\) terms:

\(\frac{D}{4} = 2abcd - a^2d^2 - b^2c^2\)

Factor out -1:

\(\frac{D}{4} = -(a^2d^2 - 2abcd + b^2c^2)\)

Recognize the expression inside the parenthesis as a perfect square: \((ad - bc)^2 = a^2d^2 - 2abcd + b^2c^2\).

So, \(\frac{D}{4} = -(ad - bc)^2\).

Condition for No Real Solutions Derived

For the quadratic equation to have no real roots, the discriminant \(D\) must be less than 0. This means \(\frac{D}{4}\) must also be less than 0.

\(-(ad - bc)^2 < 0\)

To solve this inequality, multiply both sides by -1 and reverse the inequality sign:

\((ad - bc)^2 > 0\)

For the square of a real expression, \((ad - bc)^2\), to be strictly greater than zero, the expression itself, \((ad - bc)\), must not be equal to zero. If \(ad - bc = 0\), then \((ad - bc)^2 = 0\), which is not greater than zero.

Therefore, the condition for the quadratic equation problem to have no real solutions is \(ad - bc \ne 0\).

This inequality can be written as \(ad \ne bc\). This understanding is vital for solving quadratic equations and finding real solutions in algebra and mathematics. The discriminant is a powerful tool for this analysis, providing quick math help without needing the full quadratic formula.

Conclusion and Comparison with Options

The condition for the given quadratic equation to have no real roots is \(ad \ne bc\).

Let's compare this derived condition with the given options:

1. \(ad = bc\)
2. \(ab = cd\)
3. \(ac = bd\)
4. \(ad \ne bc\)

Our derived condition, \(ad \ne bc\), matches option 4. This demonstrates how analyzing the discriminant helps identify the nature of the real solutions of a quadratic equation without full calculation.