Problem 82
Question
Find two quadratic equations having the given solutions. (There are many correct answers.) $$-2,1$$
Step-by-Step Solution
Verified Answer
The two quadratic equations having the solutions -2 and 1 are \(x^2 + x - 2 = 0\) and \(2x^2 + 2x - 4 = 0\).
1Step 1: Find the First Quadratic Equation
Given roots are -2 and 1. A quadratic equation can be written in terms of its roots as \(a(x - r_1)(x - r_2) = 0\) where \(r_1\) and \(r_2\) are roots and 'a' is a non-zero coefficient. Let's consider 'a' as 1 for the first equation. Substituting the given roots, the first equation becomes \(x^2 + x - 2 = 0\).
2Step 2: Find the Second Quadratic Equation
Now, for the second equation, let's consider 'a' as 2. Subscribe the same roots as the first equation, we get \(2(x - (-2))(x - 1) = 0\). Simplify this equation, the second equation becomes \(2x^2 + 2x - 4 = 0\).
Key Concepts
PolynomialsRoots of QuadraticAlgebraic ExpressionsFactoring Quadratics
Polynomials
Polynomials are algebraic expressions that consist of variables and coefficients, constructed using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. An example of a polynomial is the quadratic equation, which is a polynomial of degree 2, meaning the highest exponent of the variable is 2.
For instance, the quadratic equation derived from the roots -2 and 1 can be written as a polynomial in the form of \(ax^2 + bx + c = 0\). In the provided solution, the first equation \(x^2 + x - 2 = 0\) is a polynomial with coefficients \(a = 1\), \(b = 1\), and \(c = -2\).
Understanding polynomials is crucial when solving quadratic equations, since finding the roots is essentially about finding the values of 'x' for which the polynomial equals zero. Polynomials can also have different degrees, with quadratics being second-degree polynomials, characterized by the highest power of 'x' being 2.
For instance, the quadratic equation derived from the roots -2 and 1 can be written as a polynomial in the form of \(ax^2 + bx + c = 0\). In the provided solution, the first equation \(x^2 + x - 2 = 0\) is a polynomial with coefficients \(a = 1\), \(b = 1\), and \(c = -2\).
Understanding polynomials is crucial when solving quadratic equations, since finding the roots is essentially about finding the values of 'x' for which the polynomial equals zero. Polynomials can also have different degrees, with quadratics being second-degree polynomials, characterized by the highest power of 'x' being 2.
Roots of Quadratic
The roots of a quadratic equation are the values of 'x' that satisfy the equation \(ax^2 + bx + c = 0\), where 'a' is not equal to zero. These roots are also known as the solutions or zeros of the equation.
From the example provided, -2 and 1 are the solutions to the quadratic equation. We can verify these roots by substituting them back into the original equation and checking if the equation holds true, which is an essential step to ensure the accuracy of solving quadratic equations.
To find these roots, several methods can be used such as factoring, completing the square, or using the quadratic formula. The roots can be real or complex, and a quadratic equation always has exactly two roots, though they can be the same number (i.e., repeated roots).
From the example provided, -2 and 1 are the solutions to the quadratic equation. We can verify these roots by substituting them back into the original equation and checking if the equation holds true, which is an essential step to ensure the accuracy of solving quadratic equations.
To find these roots, several methods can be used such as factoring, completing the square, or using the quadratic formula. The roots can be real or complex, and a quadratic equation always has exactly two roots, though they can be the same number (i.e., repeated roots).
Algebraic Expressions
Algebraic expressions are combinations of constants, variables and arithmetic operations. A quadratic equation is a specific type of algebraic expression that takes the standard form \(ax^2 + bx + c\), where 'a', 'b', and 'c' are constants.
In the context of solving quadratic equations, the algebraic expression is manipulated to find the roots. The manipulation can include operations like factoring, expanding, and simplifying. For instance, in the second step of our provided solution, the algebraic expression \(2(x - (-2))(x - 1) = 0\) is simplified to form the quadratic equation \(2x^2 + 2x - 4 = 0\), which is an algebraic expression representing the second quadratic equation with the same solutions as the first.
In the context of solving quadratic equations, the algebraic expression is manipulated to find the roots. The manipulation can include operations like factoring, expanding, and simplifying. For instance, in the second step of our provided solution, the algebraic expression \(2(x - (-2))(x - 1) = 0\) is simplified to form the quadratic equation \(2x^2 + 2x - 4 = 0\), which is an algebraic expression representing the second quadratic equation with the same solutions as the first.
Factoring Quadratics
Factoring quadratics is a method used to solve quadratic equations by expressing them as a product of linear factors. For example, if we have a quadratic equation \(ax^2 + bx + c = 0\), factoring would involve rewriting the equation as \((x - r_1)(x - r_2) = 0\), where \(r_1\) and \(r_2\) are the roots.
In the exercise provided, the step-by-step solution shows how the given roots, -2 and 1, are used to factor the quadratic equation as \((x + 2)(x - 1)\) for the first equation, which, when expanded, gives us \(x^2 + x - 2 = 0\). Factoring is an essential skill in algebra as it provides a straightforward way to find the roots of the equation, especially when the roots are rational numbers.
In the exercise provided, the step-by-step solution shows how the given roots, -2 and 1, are used to factor the quadratic equation as \((x + 2)(x - 1)\) for the first equation, which, when expanded, gives us \(x^2 + x - 2 = 0\). Factoring is an essential skill in algebra as it provides a straightforward way to find the roots of the equation, especially when the roots are rational numbers.
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