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Chapter 9: Problem 22

Solve the equation algebraically. Check the solutions graphically. $$ 4 x^{2}=16 $$

Short Answer

Expert verified
The solutions to the equation are \(x=2\) and \(x=-2\).

Step by step solution

01

Rewrite the Equation

Firstly, it is necessary to rewrite the given equation \(4x^2=16\) in a simpler form. This can be achieved by dividing the entire equation by 4, which results in the equation \(x^2=4\).
02

Square Root

The next step is to take the square root on both sides of the equation. This will provide the potential solutions for \(x\). Remember that the square root of any number includes both a positive and a negative value, so the solutions should be \(x=2\) and \(x=-2\).
03

Check Solutions Graphically

To verify the obtained solutions graphically, plot the equation \(4x^2=16\). The intersections with the x-axis are the solutions to the equation. The graph should cross the x-axis at \(x=2\) and \(x=-2\), which is in agreement with the algebraic solution.

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Key Concepts

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

Graphical Verification
Graphical verification involves using a graph to confirm the solutions of equations. By plotting the equation, you can visually see where solutions lie. In the given equation, we had to solve for where the graph of \(4x^2 = 16\) intersects the x-axis. Here's how it works:
  • Graph the equation \(4x^2 = 16\). This is a parabola opening upwards.
  • The x-axis is where \(y=0\). Therefore, intersections indicate solutions where the equation equals zero.
  • In our case, the graph intersects the x-axis at \(x=2\) and \(x=-2\). This visually confirms our algebraic solutions.
Visual confirmation is powerful because it not only checks correctness but also enhances understanding by providing a different perspective.
Simplification of Equations
Simplification of equations involves rewriting equations to make them easier to solve. In our exercise, we simplified \(4x^2=16\) to \(x^2=4\). Simplification helps to:
  • Identify the core structure of the equation by removing unnecessary terms.
  • Make operations like solving, factoring, or graphing more straightforward.
To simplify:
  • Look for common factors. Here, we divided through by 4, resulting in \(x^2=4\).
  • Check if further simplification is possible or needed.
By simplifying early, solving processes become clearer and less error-prone.
Properties of Square Roots
Understanding the properties of square roots is crucial when solving quadratic equations. When we simplified the equation to \(x^2=4\), the next step required us to take the square root. Here are some important points:
  • The square root of a positive number results in two values: one positive and one negative. So, \(\sqrt{4} = 2\) and \(-2\).
  • Taking square roots is about finding values which when squared return the original number.
  • Square roots help us find roots of the equation. These are the solutions where the parabola intersects the x-axis.
A clear grasp of these properties helps in recognizing that equations like \(x^2=4\) have two solutions, thus ensuring accurate problem-solving.

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Most popular questions from this chapter

CHANGING C-VALUES In Exercises 21-23, find values of \(c\) so that the equation will have two solutions, one solution, and no real solution. Then sketch the graph of the equation for each value of \(c\) that you chose. $$2 x^{2}+3 x+c=0$$

LOGICAL REASONING Consider the equation \(a x^{2}+b x+c=0\) and use the quadratic formula to justify the statement. If \(b^{2}-4 a c\) is negative, then the equation has no real solution.

Use the following information. Scientists simulate a gravity-free environment called microgravity in free- fall situations. A similar microgravity environment can be felt on free-fall rides at amusement parks or when stepping off a high diving platform. The distance \(d\) (in meters) that an object that is dropped falls in \(t\) seconds can be modeled by the equation \(d=\frac{1}{2} g\left(t^{2}\right),\) where \(g\) is the acceleration due to gravity (9.8 meters per second per second). In Japan a 490 -meter-deep mine shaft has been converted into a microgravity facility. This creates the longest period of free fall currently available on Earth. How long will a period of free-fall be?

Solve the equation or write no solution. Write the solutions as integers if possible. Otherwise write them as radical expressions. $$5 x^{2}=500$$

SOLVING INEQUALITIES Solve the inequality. $$y-4 x<0$$
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