Problem 14
Question
If \(\alpha\) be a root of \(4 x^{2}+2 x-1=0\), prove that \(4 \alpha^{3}-3 \alpha\) is the other root.
Step-by-Step Solution
Verified Answer
Substituting \(4\alpha^3 - 3\alpha\) into the quadratic equation \(4x^2 + 2x - 1 = 0\), and by simplifying, it is verified that the equation holds, proving that \(4\alpha^3 - 3\alpha\) is indeed the other root of the given quadratic equation.
1Step 1: Substitution
Since \(\alpha\) is a root of the equation, it means that it can be substituted into the equation in place of \(x\), rendering it to be zero. That is, \(4\alpha^2 + 2\alpha - 1 = 0\).
2Step 2: Substitution of other root
Let's substitute \(4\alpha^3 - 3\alpha\) (going to be referred as \(\beta\)) into the quadratic equation: \(4(4\alpha^3 - 3\alpha)^2 + 2(4\alpha^3 - 3\alpha) - 1 \). Simplify this equation to get it equal to zero.
3Step 3: Verification
In this step, use the fact that \(\alpha\) is a root to the quadratic equation and therefore satisfies \(4\alpha^2 + 2\alpha - 1 = 0\). If the simplified equation from before equals to zero, then it verifies that \(4\alpha^3 - 3\alpha\) is indeed the other root.
Key Concepts
Roots of PolynomialQuadratic FormulaAlgebraic Identities
Roots of Polynomial
Understanding the roots of a polynomial is like finding the missing pieces of a puzzle. A polynomial is an expression made up of variables, coefficients, and the operations of addition, subtraction, multiplication, and non-negative integer exponents. When we're dealing with quadratic equations—specifically, polynomials of degree two—finding the roots means finding the values of the variable that make the polynomial equal to zero.
As exhibited in the exercise, if we have a quadratic equation like \(4x^2 + 2x - 1 = 0\), identifying the root \(\alpha\) means that when we substitute \(\alpha\) for \(x\), the equation balances out to zero. In algebra, these roots have specific properties and relationships. For instance, in this exercise, an interesting property is showcased where a function of a known root \(\alpha\), given as \(4\alpha^3 - 3\alpha\), can be the other root of the equation. This aligns with the concept of Viète's formulas, where the sum and products of the roots have a direct relationship with the coefficients of the polynomial.
As exhibited in the exercise, if we have a quadratic equation like \(4x^2 + 2x - 1 = 0\), identifying the root \(\alpha\) means that when we substitute \(\alpha\) for \(x\), the equation balances out to zero. In algebra, these roots have specific properties and relationships. For instance, in this exercise, an interesting property is showcased where a function of a known root \(\alpha\), given as \(4\alpha^3 - 3\alpha\), can be the other root of the equation. This aligns with the concept of Viète's formulas, where the sum and products of the roots have a direct relationship with the coefficients of the polynomial.
Quadratic Formula
The quadratic formula is a life-saver when it comes to solving quadratic equations effortlessly. It is a universal tool applicable to any quadratic equation \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(aeq 0\). The beauty of the quadratic formula is that it directly gives you the roots: \(x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\).
Using the quadratic formula can serve two purposes in the context of the exercise: firstly, to find the value of \(\alpha\), and secondly, to validate if there stands another value that, when cubed and modified to \(4\alpha^3 - 3\alpha\), serves as the other root. By plugging in the values of the coefficients from the given equation into the quadratic formula, one can derive the roots and affirm their relationship to each other and to the equation itself.
Using the quadratic formula can serve two purposes in the context of the exercise: firstly, to find the value of \(\alpha\), and secondly, to validate if there stands another value that, when cubed and modified to \(4\alpha^3 - 3\alpha\), serves as the other root. By plugging in the values of the coefficients from the given equation into the quadratic formula, one can derive the roots and affirm their relationship to each other and to the equation itself.
Algebraic Identities
Algebraic identities are like shortcuts that help simplify complex algebraic expressions quickly and accurately. They are equations that are true for all values of the variables involved. In solving quadratic equations, some of the most frequently used algebraic identities include the square of a sum or difference, the difference of squares, and the cube of a sum or difference.
For example, consider the expression \(a^3 - b^3\), which can be factored as \((a - b)(a^2 + ab + b^2)\), an algebraic identity known as the difference of cubes. This principle is a vital tool when substituting \(4\alpha^3 - 3\alpha\) as \(\beta\) into the original quadratic equation in the exercise. The ability to simplify the resulting expression by recognizing and applying the appropriate algebraic identities allows for a quicker and more efficient solution, ultimately revealing the intricate relationship between \(\alpha\) and \(\beta\) as roots of the original polynomial.
For example, consider the expression \(a^3 - b^3\), which can be factored as \((a - b)(a^2 + ab + b^2)\), an algebraic identity known as the difference of cubes. This principle is a vital tool when substituting \(4\alpha^3 - 3\alpha\) as \(\beta\) into the original quadratic equation in the exercise. The ability to simplify the resulting expression by recognizing and applying the appropriate algebraic identities allows for a quicker and more efficient solution, ultimately revealing the intricate relationship between \(\alpha\) and \(\beta\) as roots of the original polynomial.
Other exercises in this chapter
Problem 12
If the roots of the equation \(x^{2}+p x+q=0\) are obtained \(-2\) and \(-15\) when the coefficient of \(x\) was misread 17 in place of 13 , then find the corre
View solution Problem 13
Two candidates attempt to solve a quadratic of the form \(x^{2}+p x+q=0 .\) One starts with a wrong value of \(p\) and finds the roots to be 2 and 6 . The other
View solution Problem 15
Let \(a, b, c\) be real numbers with \(a \neq 0\) and let \(\alpha, \beta\) be the roots of the equation \(a x^{2}+b x+c=0 .\) Express the roots of \(a^{3} x^{2
View solution Problem 16
If \(\alpha+\beta=3\) and \(\alpha^{3}+\beta^{3}=7\), then show that \(\alpha\) and \(\beta\) are the roots of \(9 x^{2}-27 x+20=0\).
View solution