Problem 43 - Algebra 2

Question

NUMBER THEORY Use a quadratic equation to find two real numbers that satisfy each situation, or show that no such numbers exist. Their sum is 12 and their product is \(-28\)

Step-by-Step Solution

Answer

The two numbers are 14 and -2.

Step 1: Set Up the Equations

Let the two numbers be \(x\) and \(y\). We know from the problem that their sum is 12, so we write the equation: \(x + y = 12\). Similarly, their product is \(-28\), giving us the equation: \(x \cdot y = -28\).

Step 2: Express y in Terms of x

From the sum equation \(x + y = 12\), we can express \(y\) in terms of \(x\) as follows: \(y = 12 - x\).

Step 3: Substitute y in the Product Equation

Substitute \(y = 12 - x\) into the product equation \(x \cdot y = -28\). This gives us: \(x(12 - x) = -28\).

Step 4: Form a Quadratic Equation

Simplify the equation: \[x(12 - x) = -28\] which expands to \[12x - x^2 = -28\]. Rearrange this to form a standard quadratic equation: \[x^2 - 12x - 28 = 0\].

Step 5: Solve the Quadratic Equation

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to solve the equation \(x^2 - 12x - 28 = 0\) where \(a = 1, b = -12, c = -28\). Calculate the discriminant: \[b^2 - 4ac = (-12)^2 - 4(1)(-28) = 144 + 112 = 256\]. Since the discriminant is positive, there are two real solutions.

Step 6: Calculate the Roots Using the Quadratic Formula

Apply the quadratic formula: \[x = \frac{-(-12) \pm \sqrt{256}}{2(1)} = \frac{12 \pm 16}{2}\].\[x_1 = \frac{12 + 16}{2} = 14\] and \[x_2 = \frac{12 - 16}{2} = -2\]. The two numbers are \(x = 14\) and \(y = -2\), or vice versa.

Key Concepts

Number Theory

Number theory is a branch of mathematics that focuses on the properties and relationships of numbers, particularly integers. In this exercise, we apply number theoretical concepts by using a quadratic equation to find two numbers with specified conditions: their sum and product.

This task involves understanding how numbers relate to each other through addition and multiplication. Thus, it requires setting up equations that reflect these relationships based on the given conditions. Such problems typically demand skills in logical reasoning and algebra, which are foundational aspects of number theory.

• Sum of two numbers: Represented by a simple linear equation like \(x + y = 12\).
• Product of two numbers: Represented by a quadratic equation like \(x \cdot y = -28\).

These equations express fundamental number relationships within the context of quadratic solutions.

Real Numbers

Real numbers are numbers that can be found on the continuous number line and include both rational and irrational numbers. In our exercise, we need to find two real numbers whose sum is 12 and product is \(-28\).

Real numbers are an essential concept in solving practical math problems because they encompass all the measurable quantities. When two real numbers add up to 12, we are using the property of real numbers that supports such arithmetic operations. Thus, solving for them in equations reflects understanding their behavior on the number line.

• Rational numbers, like fractions and integers, are included.
• Irrational numbers, like \(\sqrt{2}\), are also encompassed.
• Real numbers are essential for constructing solutions in algebra.

This concept is central in finding solutions that exist within the field of real numbers, ensuring that results are not abstract or imaginary.

Quadratic Formula

The quadratic formula is a reliable and systematic method for solving quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\). It is expressed as \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

In this exercise, after forming the quadratic equation \(x^2 - 12x - 28 = 0\), we used the quadratic formula to determine the solutions. The discriminant \(b^2-4ac\) in the formula helps determine the nature of the solutions:

• If positive, two distinct real solutions exist.
• If zero, one real solution exists.
• If negative, no real solutions exist, but two complex ones arise.

By applying the quadratic formula, we calculated that \(x = 14\) and \(x = -2\), providing the real number solutions that satisfy both initial conditions of sum and product.

Solving Equations

Solving equations involves finding the values of variables that satisfy given mathematical statements. In this exercise, solving equations means working through algebraic manipulation to find values of \(x\) and \(y\) that satisfy the set conditions.

Simplifying equations involves re-arranging them to isolate a variable. For instance:

• Start with expressing one variable in terms of another, such as \(y = 12 - x\).
• Substitute into another equation to eliminate a variable, reducing it to one equation with one unknown.
• Use the quadratic formula when faced with a quadratic equation, as shown in this exercise.

The process employs logical step-by-step analysis and methodical calculation to find the correct solutions, which in this case are the two numbers required by the problem.