Problem 54 - Intermediate Algebra

Question

For each quadratic equation, first use the discriminant to determine whether the equation has two nonreal complex solutions, one real solution with a multiplicity of two, or two real solutions. Then solve the equation. $$ 4 x^{2}+20 x+25=0 $$

Step-by-Step Solution

Answer

One real solution: \(x = -2.5\), with multiplicity two.

Step 1: Identify the Coefficients

For the quadratic equation \(4x^2 + 20x + 25 = 0\), identify the coefficients: \(a = 4\), \(b = 20\), and \(c = 25\).

Step 2: Compute the Discriminant

The discriminant \(D\) is calculated using the formula \(D = b^2 - 4ac\). Substitute \(b = 20\), \(a = 4\), and \(c = 25\) to get: \(D = 20^2 - 4 \cdot 4 \cdot 25\).

Step 3: Simplify the Discriminant Expression

Compute the value of the discriminant: \(D = 400 - 400 = 0\).

Step 4: Interpret the Discriminant

Since \(D = 0\), the quadratic equation has one real solution with a multiplicity of two. This means the parabola touches the x-axis at one point.

Step 5: Solve the Quadratic Equation

Use the quadratic formula: \(x = \frac{-b \pm \sqrt{D}}{2a}\). Since \(D = 0\), it simplifies to \(x = \frac{-b}{2a}\). Substitute \(b = 20\) and \(a = 4\): \(x = \frac{-20}{8} = -2.5\).

Step 6: Final Step: Confirm the Solved Value

The solution for the equation \(4x^2 + 20x + 25 = 0\) is \(x = -2.5\), with a multiplicity of two.

Key Concepts

Discriminant in Quadratic Equations

The discriminant is a key element in the quadratic formula. It helps us determine the nature of the solutions for a quadratic equation. For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant \( D \) is calculated using the expression \( b^2 - 4ac \).
This tiny but mighty determinant tells us a lot about the roots of the quadratic.

• If \( D > 0 \), the equation has two distinct real solutions. The parabola will cross the x-axis at two points.
• If \( D = 0 \), there is exactly one real solution, also known as a repeated root. This means the parabola just touches the x-axis but does not cross it.
• If \( D < 0 \), the equation has two nonreal complex solutions. In this case, the parabola does not touch the x-axis at all.

Understanding the discriminant gives powerful insight into solving the equation without even finding the exact solutions yet.

Solving Quadratic Equations

Solving quadratic equations is a fundamental process in algebra. The quadratic formula provides a precise method to find the roots of any quadratic equation.
The formula is expressed as \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
First, identify the values of \( a \), \( b \), and \( c \) in your equation. Next, compute the discriminant \( D = b^2 - 4ac \).
The discriminant informs whether to proceed with further calculations for real or complex solutions.

• For \( D = 0 \), the simplified formula is \( x = \frac{-b}{2a} \). This reveals the repeated real solution.
• For \( D > 0 \), plug the values into the full formula to find the two distinct real solutions.
• For \( D < 0 \), solving proceeds with imaginary numbers, incorporating the complex roots found using \( \pm \sqrt{-D} \, i \).

The quadratic formula is a reliable tool that, paired with the discriminant, simplifies finding solutions to tricky equations.

Real and Nonreal Solutions in Quadratics

The solutions of quadratic equations can be categorized into real and nonreal (complex) solutions. This categorization hinges crucially on the discriminant's value.
Real solutions occur when the discriminant is zero or positive. If \( D = 0 \), the equation has a perfect square root, leading to one real solution with multiplicity two. This means both solutions are the same, and the graph of the quadratic touches the x-axis at one point.
When \( D > 0 \), the quadratic has two distinct real solutions. The graph will intercept the x-axis at two different points, corresponding to these solutions.
Nonreal solutions emerge when \( D < 0 \). In this case, the equation results in complex numbers since you would be taking the square root of a negative number.

• These complex roots come in conjugate pairs of the form \( a \pm bi \).
• The graph of the quadratic in the real number system will not intersect the x-axis, but in the complex plane, they have meaningful intersections.

Having a grasp of both real and nonreal solutions helps in understanding both the nature of quadratic functions and their graphical interpretations.