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Magazine Chapter 5: Problem 37
Find two numbers whose sum is 12 and whose product is 35 .
Short Answer
Expert verified
The two numbers are 7 and 5.
Step by step solution
01
Define Variables
Let the two numbers be represented as \(x\) and \(y\).
02
Set Up Equations
Based on the problem, we have two equations: 1. \(x + y = 12\) 2. \(x \times y = 35\)
03
Solve for One Variable
From the first equation, solve for \(y\): \(y = 12 - x\)
04
Substitute into the Second Equation
Substitute \(y\) from Step 3 into the second equation: \(x \times (12 - x) = 35\)
05
Solve the Quadratic Equation
Simplify and solve the equation: \(x(12 - x) = 35\) \(12x - x^2 = 35\) \(x^2 - 12x + 35 = 0\) Use the quadratic formula: \(x = \frac{-b \,\pm \, \sqrt{b^2 - 4ac}}{2a}\) Where \(a = 1\), \(b = -12\), and \(c = 35\)
06
Calculate the Discriminant
Calculate the discriminant: \(b^2 - 4ac = (-12)^2 - 4(1)(35) = 144 - 140 = 4\)
07
Find the Roots
Find the roots using the quadratic formula: \(x = \frac{12 \,\pm \, \sqrt{4}}{2}\) \(x = \frac{12 \,\pm \, 2}{2}\) \(x = 7\) or \(x = 5\)
08
Verify the Solution
Check the values of \(x\) and solve for \(y\). If \(x = 7\), \(y = 12 - 7 = 5\). If \(x = 5\), \(y = 12 - 5 = 7\).
09
Final Pair
Both solutions work since they satisfy both given conditions. The two numbers are 7 and 5.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
System of Equations
A system of equations is a set of equations with multiple variables that you need to solve simultaneously. In this exercise, our system consists of two equations that represent the sum and the product of two unknown numbers. The equations are:
1. \(x + y = 12\)
2. \(x \times y = 35\).
To solve the system, start by expressing one variable in terms of the other from one of the equations. In this case, solve the first equation for \(y\):
\(y = 12 - x\).
This substitution helps transform the system into a single quadratic equation. Solving systems of equations this way simplifies the process considerably.
1. \(x + y = 12\)
2. \(x \times y = 35\).
To solve the system, start by expressing one variable in terms of the other from one of the equations. In this case, solve the first equation for \(y\):
\(y = 12 - x\).
This substitution helps transform the system into a single quadratic equation. Solving systems of equations this way simplifies the process considerably.
Quadratic Formula
The quadratic formula is a mathematical tool used to find the roots (solutions) of a quadratic equation, which is an equation of the form \(ax^2 + bx + c = 0\). The quadratic formula is:
\(x = \frac{-b \,\pm \, \sqrt{b^2 - 4ac}}{2a}\)
In this exercise, after substituting \(y\) from the first equation into the second equation, we get the quadratic equation:
\(x^2 - 12x + 35 = 0\).
Here, \(a = 1\), \(b = -12\), and \(c = 35\). Using these coefficients in the quadratic formula, we find the solutions for \(x\). This formula is critical for solving the quadratic equation step in our problem.
\(x = \frac{-b \,\pm \, \sqrt{b^2 - 4ac}}{2a}\)
In this exercise, after substituting \(y\) from the first equation into the second equation, we get the quadratic equation:
\(x^2 - 12x + 35 = 0\).
Here, \(a = 1\), \(b = -12\), and \(c = 35\). Using these coefficients in the quadratic formula, we find the solutions for \(x\). This formula is critical for solving the quadratic equation step in our problem.
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying equations to find solutions for variables. This exercise demonstrates several algebraic steps:
Simplification techniques include combining like terms and isolating variables. Understanding these steps is crucial for effectively solving more complex problems.
- First, derive one variable in terms of the other: \(y = 12 - x\).
- Next, substitute this expression into the second equation: \(x(12 - x) = 35\).
- Then, simplify to form a quadratic equation: \(x^2 - 12x + 35 = 0\).
Simplification techniques include combining like terms and isolating variables. Understanding these steps is crucial for effectively solving more complex problems.
Discriminant Calculation
The discriminant of a quadratic equation helps determine the nature of its roots. The discriminant is given by \(b^2 - 4ac\). For our quadratic equation \(x^2 - 12x + 35 = 0\), calculate as follows:
\(b^2 - 4ac = (-12)^2 - 4(1)(35) = 144 - 140 = 4\).
Since the discriminant is positive (4 in this case), we know there are two distinct real roots. This calculation informs us about the number and type of solutions before using the quadratic formula to find the exact roots \(x = 7\) and \(x = 5\). Recognizing the importance of the discriminant helps in analyzing and predicting the nature of the roots without fully solving the equation.
\(b^2 - 4ac = (-12)^2 - 4(1)(35) = 144 - 140 = 4\).
Since the discriminant is positive (4 in this case), we know there are two distinct real roots. This calculation informs us about the number and type of solutions before using the quadratic formula to find the exact roots \(x = 7\) and \(x = 5\). Recognizing the importance of the discriminant helps in analyzing and predicting the nature of the roots without fully solving the equation.
