Problem 48 - Beginning and Intermediate Algebra

Question

Find the discriminant. Use it to determine whether the solutions for each equation are A. two rational numbers B. one rational number C. two irrational numbers D. two nonreal complex numbers. Tell whether the equation can be solved using the zero-factor property, or if the quadratic formula should be used instead. Do not actually solve. $$ 4 x^{2}-28 x+49=0 $$

Step-by-Step Solution

Answer

One rational number, solved using the zero-factor property.

Step 1: Identify the coefficients

In the quadratic equation of the form ax^2 + bx + c = 0, identify the coefficients. In this case, a = 4, b = -28, and c = 49.

Step 2: Calculate the discriminant

The discriminant Δ of the quadratic equation can be calculated using the formula: Δ = b^2 - 4ac. Substitute the values of a, b, and c: Δ = (-28)^2 - 4(4)(49).

Step 3: Simplify the discriminant

Calculate the expression: Δ = 784 - 4(4)(49) = 784 - 784 = 0.

Step 4: Determine the nature of the roots

If the discriminant (Δ) is 0, the equation has: B. one rational number .

Step 5: Determine the solving method

If the discriminant is 0 or positive, the quadratic equation can generally be solved using the zero-factor property or the quadratic formula. Since Δ is 0 and the equation has only one rational solution, the zero-factor property can be used.

Key Concepts

solving quadratic equations

A quadratic equation is an equation of the form ax^2 + bx + c = 0. To solve a quadratic equation, we primarily use two methods: the zero-factor property and the quadratic formula. The zero-factor property is used when we can factorize the quadratic equation into two binomials. On the other hand, the quadratic formula is used to find the roots directly from the coefficients. The quadratic formula is given by: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). The term inside the square root, \[b^2 - 4ac\], is known as the discriminant and it plays a crucial role in determining the nature of the solutions.

discriminant analysis

The discriminant of a quadratic equation is represented by Δ and is calculated as Δ = b^2 - 4ac. Analyzing the discriminant helps us understand the nature of the roots of the quadratic equation. Depending on the value of Δ, we can determine whether the roots are real or complex:

• If Δ > 0, there are two distinct real roots.
• If Δ = 0, there is exactly one real root, often called a repeated or double root.
• If Δ < 0, there are two complex roots.

This analysis helps us not only in understanding the type of solutions but also in deciding the solving method.

zero-factor property

The zero-factor property states that if the product of two numbers is zero, then at least one of the numbers must be zero. This property is crucial when factorizing quadratic equations. For example, if we factorize a quadratic equation such that it becomes (px + q)(rx + s) = 0, then we can set each factor to zero: px + q = 0 and rx + s = 0. Solving these linear equations gives us the roots of the quadratic equation. This method simplifies solving when the quadratic can be factorized easily.

nature of roots

Understanding the nature of the roots of a quadratic equation is essential. The roots can be real or complex, rational or irrational. Here is how the discriminant helps:
If Δ > 0, and it is a perfect square, the roots are real and rational. If Δ > 0, and it is not a perfect square, the roots are real but irrational. If Δ = 0, there is exactly one real and rational root. If Δ < 0, the roots are complex numbers and are not real. By analyzing Δ, we can quickly determine the nature of the solutions without actually solving the equation.