Question

Without solving, determine the character of the solution of the quadratic equation \(4 x^{2}-28 x+49=0\) in the complex number system.

Short answer

The quadratic equation has one real solution (a repeated root).

Step-by-step solution

- Identify the quadratic equation

The given quadratic equation is \(4 x^{2}-28 x + 49=0\).

- Recall the quadratic formula

The solutions of a quadratic equation \(a x^{2} + b x + c = 0\) are given by the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \]Here, \(a = 4\), \(b = -28\), and \(c = 49\).

- Calculate the discriminant

The discriminant \(\Delta\) of the quadratic formula is computed as: \[ \Delta = b^{2} - 4ac \]Plugging in the values, we get: \[ \Delta = (-28)^{2} - 4 \cdot 4 \cdot 49\]

- Evaluate the discriminant

Evaluate the discriminant: \[ \Delta = 784 - 784 = 0 \]So, the discriminant \(\Delta = 0\).

- Determine the character of the solution

When the discriminant is zero, the quadratic equation has exactly one real solution, which is a repeated root (double root).

Key concepts

Discriminant

The discriminant is a crucial component in determining the nature of the solutions of a quadratic equation. It is denoted by the symbol \(\backslash\textbf{Delta}\) and calculated using the formula: \[ \Delta = b^{2} - 4ac \] The values of \(a\), \(b\), and \(c\) are the coefficients from the quadratic equation \(ax^{2} + bx + c = 0\).

The discriminant reveals if the solutions are real or complex:

• If \(\backslash\textbf{Delta} > 0\), there are two distinct real solutions.
• If \(\backslash\textbf{Delta} = 0\), there is one repeated real solution.
• If \(\backslash\textbf{Delta} < 0\), the solutions are complex (conjugate pair).

For the given quadratic equation \(4x^{2} - 28x + 49 = 0\), after substituting the coefficients, we calculated the discriminant as: \[ \Delta = (-28)^{2} - 4 \times 4 \times 49 = 784 - 784 = 0 \]
Therefore, the discriminant is zero, indicating that the equation has one real solution.

Quadratic Formula

The quadratic formula provides a systematic method for solving any quadratic equation \(ax^{2} + bx + c = 0\). It is expressed as:

\[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \]
This formula includes the discriminant \(\Delta = b^{2} - 4ac\) under the square root.

• When \(\Delta > 0\), the formula yields two different real solutions.
• When \(\Delta = 0\), it simplifies to one unique solution.
• When \(\Delta < 0\), the solutions are complex with imaginary parts.

For the quadratic equation \(4x^{2} - 28x + 49 = 0\), we would normally apply the quadratic formula. However, knowing \(\Delta = 0\) simplifies our task.
The formula simplifies to: \[ x = \frac{-b}{2a} \]
which provides us the double root directly.

Repeated Root

A repeated root (or double root) occurs when a quadratic equation has exactly one solution that appears twice. This phenomenon happens when the discriminant \(\Delta\) is equal to zero.

Here's a step-by-step breakdown:

• The quadratic equation is \(ax^{2} + bx + c = 0\).
• We calculate the discriminant \(\Delta = b^{2} - 4ac\).
• If \(\Delta = 0\), the equation has a repeated root.
• The solution is found by \[ x = \frac{-b}{2a} \]

For the given equation \(4x^{2} - 28x + 49 = 0\), we have:

\[ x = \frac{-(-28)}{2 \times 4} = \frac{28}{8} = 3.5 \]
Hence, the equation has one repeated root at \(x = 3.5\). This means the parabola touches the x-axis at a single point \(x = 3.5\).