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Chapter 8: Problem 43

Find \(b^{2}-4 a c\) and the number of real solutions to each equation. $$9-24 z+16 z^{2}=0$$

Short Answer

Expert verified
The discriminant is \(0\), and there is exactly one real solution.

Step by step solution

01

- Identify coefficients

Rewrite the given equation in standard form: \[16z^2 - 24z + 9 = 0\] Identify the coefficients: \(a = 16\), \(b = -24\), \(c = 9\).
02

- Compute the discriminant

Calculate the discriminant using the formula: \(b^2 - 4ac\). Substitute the coefficients: \[(-24)^2 - 4(16)(9)\].
03

- Simplify the discriminant

Simplify the expression: \[576 - 576 = 0\]. So the discriminant is \(0\).
04

- Determine the number of real solutions

Since the discriminant is \(0\), there is exactly one real solution to the equation.

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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 to solve quadratic equations. These are equations of the form \[ ax^2 + bx + c = 0 \] where
  1. a, is the coefficient of \(x^2\)
  2. b, is the coefficient of \(x\)
  3. c, is the constant term.
The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. It helps find the 'x' values (solutions) by plugging in the values of a, b, and c from the quadratic equation into the formula. This formula is especially useful when the equation cannot be easily factored. Remember, the part under the square root, \(b^2 - 4ac\), is known as the discriminant. Understanding the quadratic formula can make solving quadratic equations much easier and faster.
real solutions
The number of real solutions of a quadratic equation is determined by the discriminant, \(b^2 - 4ac\). Here's how:
  • If \(b^2 - 4ac > 0\) - There are two distinct real solutions.
  • If \(b^2 - 4ac = 0\) - There is exactly one real solution.
  • If \(b^2 - 4ac < 0\) - There are no real solutions (only complex solutions).
In the exercise, we calculated the discriminant: \[b^2 - 4ac = 0\]. Since the discriminant is zero, the equation has exactly one real solution. Recognizing the number of real solutions helps us understand the nature of the roots without fully solving the equation.
standard form
Quadratic equations should be written in a standard form to apply the quadratic formula. The standard form of a quadratic equation is: \[ ax^2 + bx + c = 0 \]. This is important because:
  • It makes it easier to identify the coefficients a, b, and c
  • It standardizes the way we approach solving the equation
In the given exercise, the equation \(9 - 24z + 16z^2 = 0\) was rewritten in standard form \[16z^2 - 24z + 9 = 0 \]. By rearranging to this form, we can straightforwardly identify \(a = 16\), \(b = -24\), and \(c = 9\). These values are then used in the discriminant formula to find the number of real solutions. Thus, converting to standard form streamlines the solving process and helps avoid errors.

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Most popular questions from this chapter

Find the exact solution \((s)\) to each problem. If the solution(s) are irrational, then also find approximate solution(s) to the nearest tenth. More missing numbers. Find two real numbers that have a sum of 8 and a product of 2

Cooperative learning. Work with a group to write a quadratic equation that has each given pair of solutions. a) \(3+\sqrt{5}, 3-\sqrt{5}\) b) \(4-2 i, 4+2 i\) c) \(\frac{1+i \sqrt{3}}{2}, \frac{1-i \sqrt{3}}{2}\)

Solve each inequality. State the solution set using interval notation when possible. \(x^{3}+5 x^{2}-4 x-20 \geq 0\)

Solve each inequality. State the solution set using interval notation when possible. \(0.65 x^{2}+3.2 x+5.1>0\)

Find all real or imaginary solutions to each equation. Use the method of your choice. $$(x-2)^{2}=-9$$
See all solutions

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