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Chapter 1: Problem 48

Find the real solutions, if any, of each equation. Use the quadratic formula. $$ 2 x^{2}+5 x+3=0 $$

Short Answer

Expert verified
The solutions are \( x = -1 \) and \( x = -\frac{3}{2} \).

Step by step solution

01

- Identify coefficients

In the quadratic equation, compare the given equation with the standard form of a quadratic equation, which is: \[ ax^2 + bx + c = 0 \]Here, the given equation is: \[ 2x^2 + 5x + 3 = 0 \]Identify the coefficients: \( a = 2 \), \( b = 5 \), \( c = 3 \)
02

- Write down the quadratic formula

The quadratic formula is used to find the roots of a quadratic equation and is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
03

- Substitute coefficients into the quadratic formula

Substitute the coefficients \( a = 2 \), \( b = 5 \), and \( c = 3 \) into the quadratic formula: \[ x = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 2 \cdot 3}}{2 \cdot 2} \]
04

- Simplify the expression inside the square root

Calculate the value inside the square root (the discriminant): \[ 5^2 - 4 \cdot 2 \cdot 3 = 25 - 24 = 1 \]So the expression becomes: \[ x = \frac{-5 \pm \sqrt{1}}{4} \]
05

- Simplify the square root

Take the square root of 1: \[ \sqrt{1} = 1 \]So the expression now becomes: \[ x = \frac{-5 \pm 1}{4} \]
06

- Solve for the two possible values of x

There are two possible solutions for \( x \):1. \( x = \frac{-5 + 1}{4} = \frac{-4}{4} = -1 \)2. \( x = \frac{-5 - 1}{4} = \frac{-6}{4} = -\frac{3}{2} \)

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Key Concepts

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

quadratic formula
To solve a quadratic equation, like \[ 2x^2 + 5x + 3 = 0 \], we use the quadratic formula. This formula is a life-saver when it comes to finding the roots (solutions) of quadratic equations. The quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula helps us solve any quadratic equation by plugging in the coefficients from the equation. If you remember the quadratic formula, solving quadratic equations becomes much easier!
discriminant
The discriminant is a key part of the quadratic formula. It is the part under the square root, \[b^2 - 4ac\].
The value of the discriminant tells us the nature of the solutions of the quadratic equation.
  • If the discriminant is positive (like 1 in our problem), the equation has two distinct real solutions.
  • If the discriminant is zero, the equation has exactly one real solution.
  • If the discriminant is negative, the equation has no real solutions, only complex ones.
    In our example, the discriminant was calculated as 1, so we know there are two real solutions.
real solutions
Real solutions are the solutions of the equation which are real numbers, unlike complex numbers. When we solved the quadratic equation \[ 2x^2 + 5x + 3 = 0 \], we found the discriminant to be 1, which means there are two real solutions.
By simplifying the quadratic formula with our discriminant, we found the solutions to be:
  • \[ x = -1 \]
  • \[ x = -\frac{3}{2} \]
    These solutions show where the parabola represented by the equation intersects the x-axis.
coefficients
In a quadratic equation like \[ 2x^2 + 5x + 3 = 0 \], the coefficients are the numbers in front of the variables. They are crucial for solving the equation using the quadratic formula.
  • \[\text{a} = 2 \] is the coefficient of \[ x^2 \],
  • \[\text{b} = 5 \] is the coefficient of \[ x \],
  • \[\text{c} = 3 \] is the constant term.
    Working out these coefficients correctly lets us substitute them into the quadratic formula and eventually find the solution to the equation. Always double-check your coefficients before plugging them into the formula!

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Most popular questions from this chapter

The word quadratic seems to imply four (quad), yet a quadratic equation is an equation that involves a polynomial of degree \(2 .\) Investigate the origin of the term \(q u a d r a t i c\) as it is used in the expression quadratic equation. Write a brief essay on your findings.

Find the real solutions, if any, of each equation. $$ 5 a^{3}-45 a=-2 a^{2}+18 $$

Reducing the Size of a Candy Bar A jumbo chocolate bar with a rectangular shape measures 12 centimeters in length, 7 centimeters in width, and 3 centimeters in thickness. Due to escalating costs of cocoa, management decides to reduce the volume of the bar by \(10 \%\). To accomplish this reduction, management decides that the new bar should have the same 3 -centimeter thickness, but the length and width of each should be reduced an equal number of centimeters. What should be the dimensions of the new candy bar?

Elaine can complete a landscaping project in 2 hours with the help of either her husband Brian or both her two daughters. If Brian and one of his daughters work together, it would take them 4 hours to complete the project. Assuming the rate of work is constant for each person, and the two daughters work at the same rate, how long would it take Elaine, Brian, and one of their daughters to complete the project?

Discretionary Income Individuals are able to save money when their discretionary income (income after all bills are paid) exceeds their consumption. Suppose the model, \(y=-25 a^{2}+2400 a-30,700,\) represents the average discretionary income for an individual who is \(a\) years old. If the consumption model for an individual is given by \(y=160 a+7840,\) at what age is the individual able to start saving money? Round to the nearest year.
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