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Chapter 1: Problem 83

Solve each equation in Exercises \(83-108\) by the method of your choice. $$2 x^{2}-x=1$$

Short Answer

Expert verified
The solutions to the equation are \(x = 1\) and \(x = -0.5\).

Step by step solution

01

Simplify the Equation

First, we rearrange the equation to the form \(ax^{2} + bx + c = 0\). Therefore, we have \(2x^{2} - x - 1= 0\).
02

Applying the Quadratic Formula

Next, we substitute the coefficients a, b, and c, from the equation into the quadratic formula. Applying the formula, we get \(x=\frac{-(-1)卤\sqrt{(-1)^{2}-4*2*(-1)}}{2*2}\).
03

Evaluating the Expression Under the Square Root

The expression under the square root is known as the discriminant. We have \((-1)^{2}-4*2*(-1)\) which simplifies to \(9\). Now, substituting this into our quadratic formula reduces it down to \(x=\frac{1卤\sqrt{9}}{4}\).
04

Solve for x

This eventually yields \(x=\frac{1卤3}{4}\). Now, we simply solve this equation twice, once using the plus operator and once using the minus. This gives us two possible values for x: \(\frac{1+3}{4} = 1\) and \(\frac{1-3}{4} = -0.5\).

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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 provides a method to solve any quadratic equation of the form \( ax^{2} + bx + c = 0 \). When you're faced with a quadratic equation that's difficult to factorize, turning to the quadratic formula is a reliable choice. Here's the formula itself:
\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \]
After identifying the coefficients \(a\), \(b\), and \(c\) from the quadratic equation, you can insert them directly into this formula. The \(\pm\) sign means you will get two solutions which correspond to the two possible values of x that will satisfy the equation. It's an indispensable tool for students, especially when dealing with complex quadratics.
Discriminant in Quadratics
The discriminant is the part of the quadratic formula under the square root, denoted as \( b^2 - 4ac \). It tells us about the nature of the roots (solutions) without actually solving the equation.

The value of the discriminant can determine three possible scenarios:
  • If it is positive, there are two distinct real roots.
  • If it is zero, there is exactly one real root (also known as a repeated or double root).
  • If it is negative, there are no real roots but instead two complex roots.

Understanding the discriminant is key when anticipating the character of solutions and when explaining why some quadratic equations have no solution over the set of real numbers.
Simplify Equations
Before utilizing the quadratic formula or any other method to solve a quadratic equation, it's crucial to first simplify the equation. Simplifying can include expanding brackets, combining like terms, and rearranging the equation to the standard form \( ax^{2} + bx + c = 0 \).

Simplification helps to eliminate any confusion and reduces the chance of making mistakes when substituting into the quadratic formula. It also makes the task of identifying coefficients \(a\), \(b\), and \(c\) straightforward, which is essential when you're subsequently calculating the discriminant. Always take the time to simplify first to pave the way for a smoother solving process.

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Most popular questions from this chapter

Solve absolute value inequality. \(\left|\frac{3 x-3}{9}\right| \geq 1\)

This will help you prepare for the material covered in the next section. $$\text { Solve: } \frac{x+3}{4}-\frac{x-2}{3}+\frac{1}{4}$$

Solve absolute value inequality. \(|x-1| \geq 2\)

Explain why \(|x|<-4\) has no solution.

Use interval notation to express solution sets and graph each solution set on a number line. Solve linear inequality. \(\frac{4 x-3}{6}+2 \geq \frac{2 x-1}{12}\)
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