Problem 51 - 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. $$ x^{2}+4 x-21=0 $$

Step-by-Step Solution

Answer

The equation has two real solutions: \(x = 3\) and \(x = -7\).

Step 1: Calculate the Discriminant

For a quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant \(D\) is calculated using the formula \(D = b^2 - 4ac\). Here, \(a = 1\), \(b = 4\), and \(c = -21\). Substitute these values into the formula: \(D = 4^2 - 4 \times 1 \times (-21)\). Simplifying, we get \(D = 16 + 84 = 100\).

Step 2: Interpret the Discriminant

The value of the discriminant \(D = 100\) is positive. A positive discriminant indicates that the quadratic equation has two distinct real solutions.

Step 3: Solve the Quadratic Equation Using the Quadratic Formula

The quadratic formula for solving \(ax^2 + bx + c = 0\) is given by \(x = \frac{-b \pm \sqrt{D}}{2a}\). We know \(D = 100\), thus \(\sqrt{D} = \sqrt{100} = 10\). Substitute the values of \(a\), \(b\), and the square root of the discriminant into the formula: \(x = \frac{-4 \pm 10}{2 \times 1}\).

Step 4: Calculate the Solutions

Substitute into the formula to find the two solutions: 1. \(x = \frac{-4 + 10}{2} = \frac{6}{2} = 3\)2. \(x = \frac{-4 - 10}{2} = \frac{-14}{2} = -7\)The solutions are \(x = 3\) and \(x = -7\).

Key Concepts

Discriminant

In a quadratic equation, the discriminant is a key element that helps determine the nature of the solutions. It is a part of the quadratic formula, and it is calculated using the formula \( D = b^2 - 4ac \). This involves the coefficients \( a \), \( b \), and \( c \) from the equation in the form \( ax^2 + bx + c = 0 \).

The discriminant gives insight into the types of solutions the equation might have:

• If \( D > 0 \), we get two different real solutions.
• If \( D = 0 \), there is exactly one real solution with a multiplicity of two, meaning the graph of the equation touches the x-axis at one point.
• If \( D < 0 \), the solutions are two nonreal complex numbers, and the graph does not intersect the x-axis.

Understanding the discriminant is fundamental as it provides a quick assessment of how many and what type of solutions you will find without even calculating them directly.

Real Solutions

When solving quadratic equations, real solutions are those where the outcome is a set of real numbers. Real solutions mean the solutions can be plotted on the conventional number line and/or are points where the graph of the equation crosses or touches the x-axis.

These solutions can be either distinct or repeated:

• Two real and distinct solutions occur when \( D > 0 \). In such cases, the parabola representing the quadratic equation crosses the x-axis at two different points.
• A repeated real solution, typically known as a double root, happens when \( D = 0 \). Here, the vertex of the parabola touches the x-axis, and both solutions refer to the same point.

Recognizing whether a quadratic equation has real solutions is essential in context since it determines if the function's graph will intersect the x-axis.

Quadratic Formula

The quadratic formula is a universal way to find solutions to any quadratic equation, regardless of the type of solutions (real, repeated, or complex). The formula is given by:\[ x = \frac{-b \pm \sqrt{D}}{2a} \]where \( D \), the discriminant, is \( b^2 - 4ac \). To use the quadratic formula effectively, follow these steps:

• Ensure your quadratic equation is in standard form: \( ax^2 + bx + c = 0 \).
• Calculate the discriminant (\( D \)), then find the square root \( \sqrt{D} \).
• Substitute the values of \( a \), \( b \), and \( \sqrt{D} \) into the formula.
• Solve to find both values of \( x \) (if real solutions are possible).

The quadratic formula is an indispensable tool because of its general applicability to any quadratic equation, making it incredibly useful for quickly solving diverse quadratic problems.