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Chapter 10: Problem 54

Write a quadratic equation with the given solutions. \(-41\) and 5

Short Answer

Expert verified
The quadratic equation with the roots -41 and 5 is \(x^2 + 36x - 205 = 0\).

Step by step solution

01

- Recognize the Mathematical Property

Understand that the property of a quadratic equation states that if the roots of the equation \(ax^2 + bx + c = 0\) are \(p\) and \(q\), then \(p+q=-\frac{b}{a}\), and \(pq=\frac{c}{a}\). This is because in a factorized form, a quadratic equation looks like this: \(a(x - p)(x - q) = 0\).
02

- Identify Given Roots

According to the problem, the roots have been identified as -41 and 5. Hence, the equation can be expressed as \(a(x + 41)(x - 5) = 0\), which is the formula built using the mathematical property just described in step 1.
03

- Formulate the Quadratic Equation

Upon distributing through, our quadratic equation looks like this: \(a(x^2 + 36x - 205) = 0\), or \(x^2 + 36x - 205 = 0\), assuming \(a = 1\). This is the quadratic equation to represent the roots -41 and 5.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Roots of a Quadratic Equation
Finding the roots of a quadratic equation is a fundamental concept. These roots are the values of the variable, usually denoted as \( x \), that satisfy the equation. For a standard quadratic equation of the form \( ax^2 + bx + c = 0 \), the roots can be found using several methods, such as factoring, completing the square, or applying the quadratic formula.
  • The roots, also called zeros or solutions, are where the quadratic graph crosses the x-axis.
  • A quadratic can have two real roots, one real root (repeated), or two complex roots depending on the discriminant, \( b^2 - 4ac \).
When provided with specific roots, the equation can be built by reversing the factorization process. Given roots \(-41\) and \(5\), these values imply that inside the factors, you'd have \((x + 41)\) and \((x - 5)\). Expanding this will give you back the quadratic equation.
Factorization of Quadratics
Factorization is breaking down a quadratic equation into a product of its linear factors. When you know the roots of a quadratic, \(x_1\) and \(x_2\), you can express the equation as \(a(x-x_1)(x-x_2)\).
In situations where the roots are known, factorization provides a direct method to write the quadratic equation.
  • To factorize, identify the terms \((x-x_1)\) and \((x-x_2)\) for given roots.
  • Multiply these factors to convert them back into a polynomial form.
For the given roots \(-41\) and \(5\), factorization results in \(a(x + 41)(x - 5)\). Assuming \(a = 1\), when multiplied out, it simplifies to \(x^2 + 36x - 205 = 0\). This method is a fast track to obtaining the required quadratic equation.
Standard Form of a Quadratic Equation
The standard form of a quadratic equation is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients with \(a eq 0\). This format is essential because it's the most commonly used form for identifying and solving quadratic equations.
  • The coefficient \(a\) represents how "stretched" or "compressed" the parabola is.
  • The term \(b\) affects the position where the graph crosses the y-axis or the symmetry axis of the parabola.
  • The constant term \(c\) represents the y-intercept, the point where the parabola meets the y-axis.
Once roots are found or given, converting from the factorized form \((x + 41)(x - 5)\) to \(ax^2 + bx + c = 0\) involves expansion where, for the roots provided, the expanded and simplified form is \(x^2 + 36x - 205 = 0\). This step ensures the quadratic is in its standard recognizable form.

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Most popular questions from this chapter

Use factoring to solve the equation. Use a graphing calculator to check your solution if you wish. $$27-12 x^{2}=0$$

Simplify the expression. $$\frac{12 \sqrt{4}}{\sqrt{9}}$$

Use the quadratic formula to solve the equation. $$9 x^{2}-14 x-7=0$$

Factor the expression. Tell which special product factoring pattern you used. $$y^{2}+12 y+36$$

Which of the following is a correct factorization of \(72 x^{2}-24 x+2 ?\) (A) \(-9(3 x-1)^{2}\) (B) \(8\left(9 x-\frac{1}{2}\right)^{2}\) (C) \(8\left(3 x-\frac{1}{2}\right)\left(3 x-\frac{1}{2}\right)\) (D) \(-8\left(3 x-\frac{1}{2}\right)^{2}\)
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