Question

If \(a, b, c, d \in R\) then the equation \(\left(x^{2}+a x-3 b\right)\left(x^{2}-c x+b\right)\left(x^{2}-d x+2 b\right)=0\) has (1) 6 real roots (2) At least 2 real roots (3) 4 real roots (4) 3 real roots

Short answer

The equation has at least 2 real roots.

Step-by-step solution

Identify the Factors

The given equation is \((x^2 + ax - 3b)(x^2 - cx + b)(x^2 - dx + 2b) = 0\). This equation is factored into three quadratic expressions. Each quadratic can potentially have two roots.

Roots of the First Quadratic

Consider the first quadratic \(x^2 + ax - 3b = 0\). The roots can be found using the quadratic formula: \(x = \frac{-a \pm \sqrt{a^2 + 12b}}{2}\). For real roots, the discriminant \(a^2 + 12b\) must be non-negative. To ensure at least some roots, assume \(b eq 0\).

Roots of the Second Quadratic

Consider the second quadratic \(x^2 - cx + b = 0\). The roots are given by: \(x = \frac{c \pm \sqrt{c^2 - 4b}}{2}\). For real roots, the discriminant \(c^2 - 4b\) must be non-negative. Again, assume \(b eq 0\).

Roots of the Third Quadratic

Consider the third quadratic \(x^2 - dx + 2b = 0\). The roots can be found using: \(x = \frac{d \pm \sqrt{d^2 - 8b}}{2}\). For real roots, the discriminant \(d^2 - 8b\) must be non-negative. Assume \(b eq 0\).

Conclusion on the Number of Real Roots

Each quadratic equation can have 0, 1, or 2 real roots based on its discriminant. Since each of the three quadratics can independently have at least one real root assuming \(b eq 0\), there are situations where all can have real roots. At a minimum, each quadratic will have at least one real root, resulting in at least two overall real roots among the three equations.

Key concepts

Quadratic Discriminant

To start with, let's understand the quadratic discriminant. In any quadratic equation of the form \ ax^2 + bx + c = 0 \, the discriminant is given by the expression: \ b^2 - 4ac \. This discriminant helps determine the nature of the roots:

• If \(b^2 - 4ac > 0\), the equation has two distinct real roots.
• If \(b^2 - 4ac = 0\), the equation has exactly one real root, which is also called a repeated or double root.
• If \(b^2 - 4ac < 0\), the equation has no real roots, instead it has two complex conjugate roots.

For the given equation which is factored into three quadratic parts, understanding the discriminant will help us find if each part has real roots.

Real Roots

Real roots are solutions to a quadratic equation where the value of \(x\) is a real number. From earlier, we know that the discriminant's value determines whether the roots are real. Each quadratic in the given equation must have its discriminant tested to see if it results in real roots.
Consider the first quadratic equation \(x^2 + ax - 3b = 0\):

• The discriminant is \(a^2 + 12b\).
• If \(a^2 + 12b \geq 0\), the quadratic will have real roots.

Similarly, for the second quadratic \(x^2 - cx + b = 0\):

• The discriminant is \(c^2 - 4b\).
• If \(c^2 - 4b \geq 0\), the quadratic will have real roots.

Finally, for the third quadratic \(x^2 - dx + 2b = 0\):

• The discriminant is \(d^2 - 8b\).
• If \(d^2 - 8b \geq 0\), the quadratic will have real roots.

Understanding these discriminants helps in determining the minimum number of real roots in the given equation.

Quadratic Formula

The quadratic formula provides the roots of any quadratic equation \(ax^2 + bx + c = 0\). The formula is given by: \ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \ When you apply the quadratic formula to each of the equations in the given problem, you can find their respective roots:

• For \(x^2 + ax - 3b = 0\), the roots are given by: \ x = \frac{-a \pm \sqrt{a^2 + 12b}}{2} \
• For \(x^2 - cx + b = 0\), the roots are given by: \ x = \frac{c \pm \sqrt{c^2 - 4b}}{2} \
• For \(x^2 - dx + 2b = 0\), the roots are given by: \ x = \frac{d \pm \sqrt{d^2 - 8b}}{2} \

These roots are derived by plugging the coefficients into the quadratic formula and simplify as needed. Ensuring the discriminant scenarios lead us to conclude at least one real root in each quadratic, summing up potentially to a minimum of two real roots in the problem.