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Chapter 11: Problem 8

Identify \(A, B,\) and \(C\) as used in the quadratic formula. $$4 z^{2}=7$$

Short Answer

Expert verified
A = 4, B = 0, C = -7

Step by step solution

01

- Write the original equation

The given equation is: \[4z^2 = 7\]
02

- Rewrite the equation in standard quadratic form

The standard form of a quadratic equation is \[ax^2 + bx + c = 0\]. We need to rewrite the given equation in this form. First, subtract 7 from both sides: \[4z^2 - 7 = 0\]
03

- Identify the coefficients

Now compare \[4z^2 - 7 = 0\] with \[ax^2 + bx + c = 0\]. Identify the coefficients: \[A = 4, B = 0, C = -7\] Here, there is no middle term involving z, so B is 0.

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Key Concepts

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

Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations. A quadratic equation is any equation that can be written in the form:\[ax^2 + bx + c = 0\] where
  • a is the coefficient of the squared term,
  • b is the coefficient of the linear term,
  • c is the constant term.
The quadratic formula itself is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] This formula allows you to find the values of x that make the equation true. The symbol \( \pm \) means you will get two solutions: one with a plus and one with a minus.
Standard Form
To use the quadratic formula, you need to have your equation in standard form.The standard form of a quadratic equation is:\[ax^2 + bx + c = 0\]
This form is important because it clearly shows the coefficients you will need for the quadratic formula.If your equation is not in standard form, you may need to manipulate it a bit to get there.
In our given example:\[4z^2 = 7\]
First, subtract 7 from both sides to get:\[4z^2 - 7 = 0\]
Now the equation is in standard form, and you can easily identify the coefficients.
Coefficients
Coefficients are the numbers in front of the variables in an equation. In the standard form of a quadratic equation:\[ax^2 + bx + c = 0\] You will have three coefficients:
  • a is the coefficient of the \(x^2\) term (or in our case, \(z^2\)). In our example, \(a = 4\).
  • b is the coefficient of the \(x\) term. If your equation does not have an \(x\) term, like in this example, then \(b = 0\).
  • c is the constant term. In our example, \(c = -7\).
These coefficients are essential for using the quadratic formula. Identifying them correctly means solving the quadratic equation becomes straightforward.

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Most popular questions from this chapter

Suppose that a company's profit \(P,\) in thousands of dollars, on the sale of \(x\) items is given by the equation. $$P=x^{2}-6 x-40 \quad \text { where } x \geq 0$$ (a) Sketch the graph of this equation. Let \(x\) be the horizontal axis and \(P\) be the vertical axis. (b) Use the graph to determine the minimum profit the company earns. (c) How many items does the company sell if it experiences this minimum profit?

In Exercises \(1-64\), solve each of the given equations. If the equation is quadratic, use the factoring or square root method. If the equation has no real solutions, say so. $$x^{2}+4 x=32$$

Sketch the graphs of \(y=-x^{2}\) and \(y=-3 x^{2}\) on the same coordinate system. How would you describe the effect the coefficient \(-3\) has on the graph of \(y=x^{2} ?\)

Solve each of the following exercises algebraically. One leg of a right triangle is \(7 \mathrm{cm}\) long, and the hypotenuse is \(15 \mathrm{cm}\) long. What is the length of the other leg?

Solve each of the following exercises algebraically. The width of a rectangle is one third its length. If the area of the rectangle is \(20 \mathrm{sq}\) in., what are the dimensions of the rectangle?
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