Question

Solve the given quadratic inequality using the Quadratic Formula. $$ x^{2}-4 x+4 \leq 0 $$

Short answer

The solution is \(x = 2\).

Step-by-step solution

Identify the Equation

The given inequality is \(x^2 - 4x + 4 \leq 0\). We will examine the corresponding equation \(x^2 - 4x + 4 = 0\) to find its roots using the Quadratic Formula.

Set Up the Quadratic Formula

The Quadratic Formula is given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). In our case, the coefficients are \(a = 1\), \(b = -4\), and \(c = 4\).

Calculate the Discriminant

Calculate the discriminant \(b^2 - 4ac\) to determine the nature of the roots. Here \(b^2 - 4ac = (-4)^2 - 4(1)(4) = 16 - 16 = 0\). A zero discriminant indicates one real root.

Find the Root

Since the discriminant is zero, the quadratic equation has one repeated root. Using the Quadratic Formula \(x = \frac{-(-4) \pm 0}{2(1)} = \frac{4}{2} = 2\). Hence, the root is \(x = 2\).

Write the Solution for the Inequality

The inequality \(x^2 - 4x + 4 \leq 0\) can be written as \((x - 2)^2 \leq 0\). Since \((x - 2)^2\) is always non-negative, the inequality is satisfied only when \((x - 2)^2 = 0\). Thus, the solution is \(x = 2\).

Key concepts

Quadratic Formula

The quadratic formula is a reliable method for finding the roots of any quadratic equation. A quadratic equation is expressed as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. The formula to find the roots is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This powerful tool allows you to determine the values of \(x\) that satisfy the equation.

For the quadratic equation \(x^2 - 4x + 4 = 0\), identifying \(a = 1\), \(b = -4\), and \(c = 4\) enables you to substitute these values into the formula. This substitution will set you up to calculate the discriminant next, leading you to find the roots of the equation. The quadratic formula is especially useful because it applies universally to all quadratic equations, providing a straightforward path to finding solutions.

Discriminant

The discriminant is a component of the quadratic formula and plays a crucial role in understanding the nature of the roots of a quadratic equation. It is expressed as \(b^2 - 4ac\). Depending on its value, the discriminant tells us how many real roots the equation has.

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If it is zero, the equation has one real root, or a repeated root.
• If it is negative, there are no real roots, but instead two complex roots.

In this exercise, the discriminant is \(0\). Therefore, the quadratic equation \(x^2 - 4x + 4 = 0\) has one repeated real root. Understanding the discriminant helps you predict the number of real solutions before actually solving the equation.

Real Root

The concept of a real root is central when discussing the solutions to a quadratic equation. When the discriminant equals zero, as it does in the equation \(x^2 - 4x + 4 = 0\), it indicates that the equation has exactly one real root, often called a repeated root.

In such cases, the root can be calculated directly using the quadratic formula. For \(x^2 - 4x + 4 = 0\), using the formula gives us \(x = \frac{-(-4) \pm 0}{2(1)} = 2\). This means the quadratic touches or just skims the x-axis at one point, \(x = 2\). Echolocation of a real root like this helps in visualizing the graph of a quadratic expression as a parabola, knowing exactly where it meets the x-axis.

Inequality Solution

Solving quadratic inequalities involves finding the set of values of \(x\) that satisfy the inequality condition rather than just equality. Here, the inequality \(x^2 - 4x + 4 \leq 0\) can be transformed to the equivalent expression \((x-2)^2 \leq 0\).

Notice that \((x-2)^2\) is always non-negative because it is a perfect square; it can only equal zero. Thus, the solution to the inequality can only occur when the expression is exactly zero, leading to \(x = 2\).

In other words, the inequality \(x^2 - 4x + 4 \leq 0\) holds true only at this specific point. Finding solutions to inequalities often requires examining whether a quadratic expression is greater than, less than, or equal to zero, helping you understand how the range of possible values behaves on the number line.