/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 94 Solve \(2 x^{2}-3 x+1=0\)... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 94

Solve \(2 x^{2}-3 x+1=0\)

Short Answer

Expert verified
The solutions are \(x = 1\) and \(x = \frac{1}{2}\).

Step by step solution

01

- Identify the coefficients

First, identify the coefficients from the quadratic equation of the form \(ax^2 + bx + c = 0\). In this equation, \(a = 2\), \(b = -3\), and \(c = 1\).
02

- Write down the quadratic formula

Recall the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. In this formula, substitute the coefficients \(a\), \(b\), and \(c\) identified in Step 1.
03

- Substitute the coefficients into the formula

Substitute \(a = 2\), \(b = -3\), and \(c = 1\) into the quadratic formula: \[x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(2)(1)}}{2(2)}\].
04

- Simplify under the square root

Simplify the expression under the square root: \[x = \frac{3 \pm \sqrt{9 - 8}}{4}\].
05

- Simplify the square root and numerator

Further simplify the expression: \[x = \frac{3 \pm 1}{4}\].
06

- Solve for the two possible values of x

Finally, solve for the two possible values of \(x\): \[x = \frac{3 + 1}{4} = 1\] and \[x = \frac{3 - 1}{4} = \frac{1}{2}\].

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Key Concepts

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

quadratic formula
To solve quadratic equations, we often use the quadratic formula. This is a reliable method to find the solutions for any quadratic equation of the form \(ax^2 + bx + c = 0\). The quadratic formula is:
\ \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \ \]
Here:
  • \
roots of quadratic equations
The solutions to a quadratic equation are often referred to as the roots. These roots represent the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). With the quadratic formula:

\ \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \ \], we can find one or two roots based on the discriminant (the part under the square root).
Let's take a closer look at how we use the formula:

1. Plug in the coefficients \(a\), \(b\), and \(c\) from the quadratic equation into the formula.
2...
discriminant
The discriminant is a crucial part of the quadratic equation. It helps determine the nature and number of roots. The discriminant is the term \(b^2 - 4ac\) found under the square root in the quadratic formula.
Depending on the value of the discriminant:
  • If \(b^2 - 4ac > 0\), the quadratic equation has two distinct real roots.
  • If \(b^2 - 4ac = 0\), the quadratic equation has exactly one real root, sometimes called a repeated root.
  • If \(b^2 - 4ac < 0\), the quadratic equation has no real roots; instead, it has two complex roots.
In t...

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