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Chapter 3: Problem 10

Solve. $$2 x^{2}-20=0$$

Short Answer

Expert verified
x = \pm \sqrt{10}

Step by step solution

01

Identify and Set Up the Equation

The given equation is in the standard form of a quadratic equation:\[ 2x^2 - 20 = 0 \]
02

Move the Constant Term

Add 20 to both sides of the equation to isolate the quadratic term:\[ 2x^2 = 20 \]
03

Divide by the Coefficient of x^2

Divide both sides of the equation by 2 to further simplify:\[ x^2 = 10 \]
04

Take the Square Root

Take the square root of both sides to solve for x:\[ x = \pm \sqrt{10} \]

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Key Concepts

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

solving quadratic equations
Solving quadratic equations involves finding the values of the variable that make the equation true. The standard form of a quadratic equation is typically written as \[ ax^2 + bx + c = 0 \] Here are the common steps:
  • First, you need to rearrange the equation, if necessary, to ensure it's in the standard form.
  • Isolate the quadratic term.
  • Divide by the coefficient of the quadratic term, if applicable.
  • Take the square root of both sides of the equation (if necessary).
  • Apply either factoring, completing the square, or the quadratic formula to find the solutions.
In our example, solving \[ 2x^2 - 20 = 0 \], the steps taken were isolating and solving for \( x \).
square roots
Understanding square roots is essential when dealing with quadratic equations. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because \[ 3 \times 3 = 9 \]. Some important points to remember:
  • Square roots are represented using the symbol \( \sqrt{ } \).
  • Any positive number has two square roots: a positive and a negative root.
  • For example, \( \sqrt{10} \) represents \( \pm \sqrt{10} \).
In our equation \[ x^{2} = 10 \], solving for \( x \) involves taking the square root of both sides, leading us to \[ x = \pm \sqrt{10} \].
standard form of a quadratic equation
The standard form of a quadratic equation is crucial for solving. It is expressed as: \[ ax^2 + bx + c = 0 \] Here, \( a \), \( b \), and \( c \) are coefficients, and \( x \) is the unknown variable. A few key points to note:
  • The term \( ax^2 \) is called the quadratic term.
  • The term \( bx \) is called the linear term.
  • The term \( c \) is called the constant term.
In our example, \[2 x^2 - 20 = 0\], we see the quadratic term is \(2 x^2 \), there's no linear term (\(b=0\)) and the constant term is \(-20\). Rearranging or manipulating such equations first by isolating terms helps in solving them effectively.

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Most popular questions from this chapter

A model rocket is launched with an initial velocity of \(120 \mathrm{ft} / \mathrm{sec}\) from a height of \(80 \mathrm{ft}\). The height of the rocket, in feet, \(t\) seconds after it has been launched is given by the function \(s(t)=-16 t^{2}+120 t+80 .\) Determine the time at which the rocket reaches its maximum height and find the maximum height.

Determine whether the statement is true or false. The sum of two numbers that are complex conjugates of each other is always a real number.

Use a graphing calculator to find the zeros of the function. Round to three decimal places. $$f(x)=1.21 x^{2}-2.34 x-5.63$$

a) Find the vertex. b) Determine whether there is a maximum or a minimum value and find that value. c) Find the range. d) Find the intervals on which the function is increasing and the intervals on which the function is decreasing. $$f(x)=\frac{1}{2} x^{2}-3 x+\frac{5}{2}$$

a) Find the vertex. b) Find the axis of symmetry. c) Determine whether there is a maximum or a minimum value and find that value. d) Graph the function. $$f(x)=x^{2}-8 x+12$$
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