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Chapter 1: Problem 81

Find the real solutions, if any, of each equation. Use any method. $$ 16 x^{2}-8 x+1=0 $$

Short Answer

Expert verified
The only solution is \( x = \frac{1}{4} \).

Step by step solution

01

Recognize the type of equation

The given equation is a quadratic equation in the form of \( ax^2 + bx + c = 0 \). Here, \( a = 16 \), \( b = -8 \), and \( c = 1 \).
02

Use the quadratic formula

The quadratic formula is given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substitute \( a = 16 \), \( b = -8 \), and \( c = 1 \) into the formula.
03

Calculate the discriminant

The discriminant is \( b^2 - 4ac \). So, calculate \( (-8)^2 - 4 \cdot 16 \cdot 1 = 64 - 64 = 0 \).
04

Solve for x

Since the discriminant is 0, the quadratic equation has exactly one real solution. Using \( x = \frac{-(-8) \pm \sqrt{0}}{2 \cdot 16} \), we get \( x = \frac{8}{32} = \frac{1}{4} \).

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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 an essential tool for solving quadratic equations like \( ax^2 + bx + c = 0 \). It's expressed as \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). By plugging the values of \( a \), \( b \), and \( c \) into this formula, you can find the solutions, or roots, of the quadratic equation.
Let's understand this more clearly with an example from the exercise:
For \( 16x^2 - 8x + 1 = 0 \), the values are \( a = 16 \), \( b = -8 \), and \( c = 1 \). When these values are substituted into the quadratic formula, it will help find the real solutions of the equation.
Discriminant
The discriminant is a key part of the quadratic formula and is found under the square root: \( b^2 - 4ac \). The value of the discriminant helps determine the nature of the roots of the quadratic equation:
  • If the discriminant is positive (\( b^2 - 4ac > 0 \)), the equation has two distinct real roots.
  • If the discriminant is zero (\( b^2 - 4ac = 0 \)), the equation has exactly one real root (or a repeated root).
  • If the discriminant is negative (\( b^2 - 4ac < 0 \)), the equation has no real roots, only complex ones.

In our exercise's example, for the equation \( 16x^2 - 8x + 1 = 0 \), we calculate the discriminant as: \( (-8)^2 - 4 \times 16 \times 1 = 64 - 64 = 0 \). Since the discriminant is zero, this indicates that the equation has exactly one real solution.
Real Solutions
Finding real solutions involves determining which values of \( x \) satisfy the quadratic equation. Using the quadratic formula, solving for \( x \) while considering the discriminant helps achieve this.
In our specific example, since the discriminant is 0, we use the simplified quadratic formula version: \( x = \frac{-b}{2a} \). Plugging in the values:
\( x = \frac{-(-8)}{2 \times 16} = \frac{8}{32} = \frac{1}{4} \).
Therefore, the real solution for the equation \( 16x^2 - 8x + 1 = 0 \) is \( x = \frac{1}{4} \). This is a single, unique real solution, a property indicated by the zero discriminant.

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Most popular questions from this chapter

A bathroom tub will fill in 15 minutes with both faucets open and the stopper in place. With both faucets closed and the stopper removed, the tub will empty in 20 minutes. How long will it take for the tub to fill if both faucets are open and the stopper is removed?

In the 2016 Olympics, Usain Bolt of Jamaica won the gold medal in the 100 -meter race with a time of 9.81 seconds. In the 1896 Olympics, Thomas Burke of the United States won the gold medal in the 100-meter race in 12.0 seconds. If they ran in the same race, repeating their respective times, by how many meters would Bolt beat Burke?

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