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Chapter 1: Problem 89

Find the real solutions, if any, of each equation. Use any method. $$ x^{2}+x=4 $$

Short Answer

Expert verified
\( x = \frac{-1 + \sqrt{17}}{2} \) or \( x = \frac{-1 - \sqrt{17}}{2} \)

Step by step solution

01

- Move All Terms to One Side

Start by moving all terms to one side to set the equation to zero. Subtract 4 from both sides:\[ x^2 + x - 4 = 0 \]
02

- Identify Coefficients for Quadratic Formula

The equation is now in the standard form \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = 1 \), and \( c = -4 \).
03

- Apply the Quadratic Formula

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) with the identified coefficients.\[ x = \frac{-(1) \pm \sqrt{(1)^2 - 4(1)(-4)}}{2(1)} \]Simplify inside the square root:\[ x = \frac{-1 \pm \sqrt{1 + 16}}{2} = \frac{-1 \pm \sqrt{17}}{2} \]
04

- Simplify the Two Solutions

Finally, solve for the two possible values of \( x \):\[ x = \frac{-1 + \sqrt{17}}{2} \]and\[ x = \frac{-1 - \sqrt{17}}{2} \]

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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. It provides a direct way to find the solutions. The quadratic formula is given by: ' x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ' Here: \[ -b \text{ represents the term that changes the sign of } b \] \[ \sqrt{b^2 - 4ac} \text{ represents the discriminant } \] \[ 2a \text{ represents the division by double the leading coefficient} \] The '卤' symbol means there will generally be two solutions: one for the addition and one for the subtraction. This way, the quadratic formula captures both possible solutions of the quadratic equation.
Standard Form of a Quadratic Equation
To use the quadratic formula, the quadratic equation must be in its standard form. The standard form of a quadratic equation looks like this: ' ax^2 + bx + c = 0 ' This form has three parts: \[ a \text{ is the coefficient of } x^2 \] \[ b \text{ is the coefficient of } x \] \[ c \text{ is the constant term} \] By setting the equation to zero, we can identify the coefficients necessary to use the quadratic formula. In our exercise, the original equation was 'x^{2} + x = 4'. To set it to zero, we moved the 4 to the left side, resulting in 'x^2 + x - 4 = 0'.
Identifying Coefficients
When you have the equation in standard form, the coefficients can be easily identified. These coefficients are critical for applying the quadratic formula. In the equation ' x^2 + x - 4 = 0 ': \[ a = 1 \text{ (coefficient of } x^2 \text{) } \] \[ b = 1 \text{ (coefficient of } x \text{) } \] \[ c = -4 \text{ (constant term)} \] Once these coefficients are identified, they can be plugged into the quadratic formula to find the solutions. In summary, proper identification and arrangement of these coefficients play a crucial role in solving the quadratic equation accurately!

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Most popular questions from this chapter

Find the real solutions of each equation. Use a calculator to express any solutions rounded to two decimal places. $$ x-4 x^{1 / 2}+2=0 $$

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 going from Chicago to Atlanta, a car averages 45 miles per hour, and in going from Atlanta to Miami, it averages 55 miles per hour. If Atlanta is halfway between Chicago and Miami, what is the average speed from Chicago to Miami? Discuss an intuitive solution. Write a paragraph defending your intuitive solution. Then solve the problem algebraically. Is your intuitive solution the same as the algebraic one? If not, find the flaw.

Dimensions of a Patio A contractor orders 8 cubic yards of premixed cement, all of which is to be used to pour a patio that will be 4 inches thick. If the length of the patio is specified to be twice the width, what will be the patio dimensions? (1 cubic yard = 27 cubic feet)

The distance to the surface of the water in a well can sometimes be found by dropping an object into the well and measuring the time elapsed until a sound is heard. If \(t_{1}\) is the time (measured in seconds) that it takes for the object to strike the water, then \(t_{1}\) will obey the equation \(s=16 t_{1}^{2}\), where \(s\) is the distance (measured in feet). It follows that \(t_{1}=\frac{\sqrt{s}}{4}\). Suppose that \(t_{2}\) is the time that it takes for the sound of the impact to reach your ears. Because sound waves are known to travel at a speed of approximately 1100 feet per second, the time \(t_{2}\) to travel the distance \(s\) will be \(t_{2}=\frac{s}{1100} .\) See the illustration. Now \(t_{1}+t_{2}\) is the total time that elapses from the moment that the object is dropped to the moment that a sound is heard. We have the equation $$ \text { Total time elapsed }=\frac{\sqrt{s}}{4}+\frac{s}{1100} $$ Find the distance to the water's surface if the total time elapsed from dropping a rock to hearing it hit water is 4 seconds.
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