Problem 25 - Intermediate Algebra

Question

Solve each inequality. $$ 4 x^{2}+20 x+25 \leq 0 $$

Step-by-Step Solution

Answer

The solution is \(x = -2.5\).

Step 1: Recognize the Quadratic Form

The given inequality is a quadratic inequality of the form \(ax^2 + bx + c \leq 0\), where \(a = 4\), \(b = 20\), and \(c = 25\). We need to solve it to find the range of \(x\) values for which this condition holds.

Step 2: Find the Discriminant

Evaluate the discriminant \(D = b^2 - 4ac\) to determine the nature of the roots. For our quadratic: \[D = 20^2 - 4 \times 4 \times 25 = 400 - 400 = 0\]. A zero discriminant means that there is exactly one real root, suggesting that the parabola touches the x-axis at one point.

Step 3: Find the Double Root

The double root of the quadratic equation \(4x^2 + 20x + 25 = 0\) can be found using the formula \(x = \frac{-b}{2a}\). Substituting the values, we get: \[x = \frac{-20}{2 \times 4} = \frac{-20}{8} = -2.5\].

Key Concepts

Understanding the Discriminant

In the world of quadratic equations, the discriminant helps you to know how many real roots an equation has. The discriminant is calculated using the formula \(D = b^2 - 4ac\), where \(a\), \(b\), and \(c\) are coefficients of the quadratic equation \(ax^2 + bx + c = 0\). The size and sign of \(D\) tell us:

• If \(D > 0\), there are two distinct real roots.
• If \(D = 0\), there is exactly one real root, also known as a double root.
• If \(D < 0\), there are no real roots (the roots are complex or imaginary).

In our exercise, the calculated discriminant was \(D = 0\). This means the quadratic inequality has only one solution or root. This makes our work a bit easier, as we only need to focus on this singular point.

Real Roots Clarified

Real roots are solutions to the quadratic equation that lie on the real number line. When dealing with quadratic inequalities, real roots are crucial because they help determine the intervals of \(x\) that satisfy the inequality. For example, if an equation has two real roots, these roots divide the number line into different regions. We can then test these regions to see where the inequality holds true. In the current exercise, the discriminant being zero means the quadratic equation has one real root, making it straightforward as there is only one critical point to examine. This happens at \(x = -2.5\).With just one real root, the parabola just touches the x-axis, indicating that the inequality \(4x^2 + 20x + 25 \leq 0\) is satisfied only at \(x = -2.5\).

An Exploration of Double Root

A double root occurs when a quadratic equation touches the x-axis at just one point instead of crossing it twice. This happens when the discriminant \(D = 0\). In terms of function graphing, this means the graph of the equation is tangent to the x-axis. In our exercise, the calculation shows that the double root is \(x = -2.5\). This root signifies the vertex of the parabola formed by the quadratic equation. Hence, it's the only real solution to the original inequality \(4x^2 + 20x + 25 \leq 0\).Being aware of when a double root arises helps:

• To predict parabola behavior without calculating points needlessly.
• To quickly assess where inequalities hold, especially in cases like ours where other potential solutions might not exist.

Understanding double roots simplifies the process of evaluating quadratic inequalities greatly!