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Chapter 0: Problem 105

Write a quadratic equation in standard form that has the solution set \(\\{2,5\\} .\) Alternate solutions are possible.

Short Answer

Expert verified
The quadratic equation is \(x^2 - 7x + 10 = 0\).

Step by step solution

01

Understand the Solutions

The problem gives us two solutions for a quadratic equation: \(2\) and \(5\). These solutions correspond to the values of \(x\) where the quadratic expression equals zero.
02

Form Factor Expression

For a quadratic equation with solutions \(x=2\) and \(x=5\), the factors of the equation are \((x-2)\) and \((x-5)\). A quadratic equation can be expressed in the form \((x-a)(x-b)\), where \(a\) and \(b\) are the roots.
03

Expand the Expression

Expand the expression \((x-2)(x-5)\). Use the distributive property (also known as FOIL method for these types of expressions): \((x-2)(x-5) = x^2 - 5x - 2x + 10\).
04

Simplify the Expression

Combine like terms in the expanded equation: \(x^2 - 5x - 2x + 10 = x^2 - 7x + 10\). This is the quadratic equation in its expanded standard form.
05

Write the Quadratic Equation

The standard form of the quadratic equation based on the given solutions is \(x^2 - 7x + 10 = 0\).

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Key Concepts

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

Standard Form
The standard form of a quadratic equation is one of the most important concepts in algebra. It provides a structured way to express quadratic equations, which makes them easier to analyze and solve. The general form of a quadratic equation is given by:\( ax^2 + bx + c = 0 \),where:
  • \( a \), \( b \), and \( c \) are coefficients, with \( a eq 0 \) to ensure the equation is indeed quadratic.
  • \( x \) represents the variable.
  • \( c \) is a constant term.
This format is beneficial because it allows you to easily identify key features of the quadratic such as its vertex, axis of symmetry, and more. When presented in standard form, quadratics can be tackled using various methods like factoring, completing the square, or using the quadratic formula.
Roots of Quadratic
Finding the roots of a quadratic equation is synonymous with solving the equation. The roots are the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). These roots are also referred to as solutions or zeros. In the context of graphing, they are the \( x \)-intercepts where the parabola touches or crosses the horizontal axis.For instance, if we know the roots of a quadratic are 2 and 5, the equation can be expressed as \((x-2)(x-5)=0\). Each root corresponds to a factor:
  • The root \( x=2 \) gives the factor \((x-2)\).
  • The root \( x=5 \) gives the factor \((x-5)\).
If a quadratic has equal roots, the equation may look like this \( (x-r)(x-r) \). Thus, understanding roots is critical for rewriting quadratic equations and solving them effectively.
Factoring
Factoring is a method used to solve quadratic equations by expressing them as a product of their linear factors. This is particularly helpful when the quadratic can be easily broken down into factors without complex calculations. The process involves:
  • Identifying the roots of the quadratic.
  • Writing the equation as \((x-r_1)(x-r_2)\), where \(r_1\) and \(r_2\) are the roots of the equation.
For example, let's factorize \( x^2 - 7x + 10 \), which represents a quadratic equation with roots 2 and 5. Using these roots, we write the equation as \((x-2)(x-5)\). The factoring method is particularly powerful because it can quickly simplify a problem, reveal the solutions, and help in switching back to standard form if needed. It also offers an intuitive understanding of what the roots represent in terms of the equation.

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Most popular questions from this chapter

Explain the mistake that is made. Solve the equation \(4 x+3=6 x-7\) Solution: Subtract \(4 x\) and add 7 to the equation. Divide by 3 This is incorrect. What mistake was made? \(3=6 x\) \(x=2\)

Graph the function represented by each side of the equation in the same viewing rectangle and solve for \(x\) $$3(x+2)-5 x=3 x-4$$

Solve the equation \(\sqrt{x-3}=4-\sqrt{x+2} .\) Plot both sides of the equation in the same viewing screen, \(y_{1}=\sqrt{x-3}\) and \(y_{2}=4-\sqrt{x+2},\) and zoom in on the \(x\) -coordinate of the point of intersection. Does the graph agree with your solution?

Refer to the following: Einstein's special theory of relativity states that time is relative: Time speeds up or slows down, depending on how fast one object is moving with respect to another. For example, a space probe traveling at a velocity \(v\) near the speed of light \(c\) will have "clocked" a time \(t\) hours, but for a stationary observer on Earth that corresponds to a time \(t_{0} .\) The formula governing this relativity is given by $$ t=t_{0} \sqrt{1-\frac{v^{2}}{c^{2}}} $$ If the time elapsed on a space probe mission is 5 years but the time elapsed on Earth during that mission is 30 years, how fast is the space probe traveling? Give your answer relative to the speed of light.

Find an equation of a line that passes through the point \((-A, B-1)\) and is perpendicular to the line \(A x+B y=C\) Assume that \(A\) and \(B\) are both nonzero.
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