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Chapter 1: Problem 38

Find all real solutions of the equation. $$2 x^{2}+3 x-\frac{1}{2}=0$$

Short Answer

Expert verified
The solutions are \( x = \frac{-3 + \sqrt{13}}{4} \) and \( x = \frac{-3 - \sqrt{13}}{4} \).

Step by step solution

01

Identify the Type of Equation

The given equation is a quadratic equation of the form \( ax^2 + bx + c = 0 \). Comparatively, we have \( a = 2 \), \( b = 3 \), and \( c = -\frac{1}{2} \). The task is to find the values of \( x \) that satisfy this equation.
02

Use the Quadratic Formula

The quadratic formula is used to find the roots of a quadratic equation \( ax^2 + bx + c = 0 \). It is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Substituting the values, \( a = 2 \), \( b = 3 \), and \( c = -\frac{1}{2} \) into the formula.
03

Calculate the Discriminant

Calculate the discriminant \( \Delta = b^2 - 4ac \). Here,\[\Delta = 3^2 - 4 \times 2 \times \left(-\frac{1}{2}\right) = 9 + 4 = 13\]Since the discriminant is positive, there will be two distinct real roots.
04

Find the Solutions

Substitute back into the quadratic formula:\[x = \frac{-3 \pm \sqrt{13}}{4}\]This results in two solutions:1. \( x_1 = \frac{-3 + \sqrt{13}}{4} \)2. \( x_2 = \frac{-3 - \sqrt{13}}{4} \)

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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 a powerful tool for solving quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). The formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula allows us to find the values of \( x \) that satisfy the equation by substituting the coefficients \( a \), \( b \), and \( c \) into it. When looking at the equation \( 2x^2 + 3x - \frac{1}{2} = 0 \), the coefficients are \( a = 2 \), \( b = 3 \), and \( c = -\frac{1}{2} \).
  • The \( -b \) term changes the sign of \( b \), which is the linear coefficient.
  • The \( \pm \sqrt{b^2 - 4ac} \) represents the two possible solutions, caused by the positive and negative square root.
  • The entire expression is divided by \( 2a \), which accounts for the parabolic shape of the quadratic equation.
This formula provides two solutions for \( x \), which are the points where the parabola intersects the x-axis, or the solutions for the quadratic equation.
Discriminant
The discriminant is the part of the quadratic formula under the square root symbol, \( b^2 - 4ac \), and is key in determining the nature of the roots of a quadratic equation. In our equation, \( 2x^2 + 3x - \frac{1}{2} = 0 \), the discriminant is calculated as:\[\Delta = 3^2 - 4 \times 2 \times \left(-\frac{1}{2}\right)\]When calculating this, we find:\[\Delta = 9 + 4 = 13\]
  • If \( \Delta > 0 \), as in our case, the equation has two distinct real solutions.
  • If \( \Delta = 0 \), the equation has exactly one real solution (a repeated root).
  • If \( \Delta < 0 \), the equation has no real solutions, but two complex solutions instead.
Understanding the discriminant helps us know in advance the number and type of solutions we should expect for a given quadratic equation.
Real Solutions
The solutions to a quadratic equation are often called the roots of the equation. In the context of the discriminant, we can determine the nature of these roots, whether they are real or complex. For the quadratic equation \( 2x^2 + 3x - \frac{1}{2} = 0 \), the discriminant \( \Delta = 13 \) is greater than zero, which means we have two distinct real solutions.Using the quadratic formula:\[x = \frac{-3 \pm \sqrt{13}}{4}\]We get the two solutions as:1. \( x_1 = \frac{-3 + \sqrt{13}}{4} \)2. \( x_2 = \frac{-3 - \sqrt{13}}{4} \)
  • Real solutions are the x-values where the parabola described by the quadratic equation crosses the x-axis.
  • Having two distinct real solutions indicates the parabola intersects the x-axis at two points.
  • These values are found through the calculation from the quadratic formula, taking into account the positive and negative scenarios of the square root in the formula.

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Most popular questions from this chapter

Use the formula \(h=-16 t^{2}+v_{0} t\) discussed in Example 7. A ball is thrown straight upward at an initial speed of \(v_{0}=40 \mathrm{ft} / \mathrm{s}\). a. When does the ball reach a height of \(24 \mathrm{ft} ?\) b. When does it reach a height of \(48 \mathrm{ft} ?\) c. What is the greatest height reached by the ball? d. When does the ball reach the highest point of its path? e. When does the ball hit the ground?

The Quadratic Formula gives us the roots of a quadratic equation from its coefficients. We can also obtain the coefficients from the roots. For example, find the roots of the equation \(x^{2}-9 x+20=0\) and show that the product of the roots is the constant term 20 and the sum of the roots is 9 , the negative of the coefficient of \(x\). Show that the same relationship between roots and coefficients holds for the following equations: $$\begin{aligned} &x^{2}-2 x-8=0\\\&x^{2}+4 x+2=0\end{aligned}$$ Use the Quadratic Formula to prove that in general, if the equation \(x^{2}+b x+c=0\) has roots \(r_{1}\) and \(r_{2},\) then \(c=r_{1} r_{2}\) and \(b=-\left(r_{1}+r_{2}\right)\)

Use Table 2 to help write the following numbers. 1.87 megawatts using scientific notation

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The approximate chance of winning the Powerball lottery jackpot can be expressed as \(5.8 E-9\) Write this number in decimal form.
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