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Chapter 2: Problem 5

Use the quadratic formula to solve the equation. $$x^{2}-5 x+3=0$$

Short Answer

Expert verified
The solutions to the equation are \(x = \frac{5+\sqrt{13}}{2}\) and \(x = \frac{5-\sqrt{13}}{2}\).

Step by step solution

01

Identifying coefficients

First, identify the coefficients \(a\), \(b\), and \(c\) in the given equation \(x^{2}-5 x+3=0\). Here, \(a=1\), \(b=-5\), and \(c=3\).
02

Calculate the discriminant

Next, calculate the discriminant, which is \(b^2 - 4ac\). Thus, the discriminant is \((-5)^2 - 4*1*3 = 25 - 12 = 13\).
03

Substitute into the quadratic formula

Now, substitute \(a=1\), \(b=-5\), and \(c=3\) into the quadratic formula. This gives \(x = \frac{-(-5) \pm \sqrt{13}}{2*1}\).
04

Calculate the roots

Finally, calculate the roots by simplifying the above expression. This gives the two solutions: \(x = \frac{5+\sqrt{13}}{2}\) and \(x = \frac{5-\sqrt{13}}{2}\).

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Key Concepts

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

Understanding the Discriminant
The discriminant is a key component of the quadratic formula, used to determine the nature of the roots of a quadratic equation. It is represented by the expression \( b^2 - 4ac \). Here's how it works:
  • If the discriminant is positive, the quadratic equation has two distinct real roots.
  • If the discriminant is zero, the equation has exactly one real root (also known as a repeated root).
  • If the discriminant is negative, the quadratic equation has two complex roots (which are not real).
The discriminant essentially gives a quick insight into the type of solutions we can expect without actually solving for the roots. In our example, with the equation \( x^{2}-5x+3=0 \), the discriminant is calculated as \( 13 \), which is positive. This indicates that the equation has two distinct real roots. Knowing how to interpret the discriminant can save time and help verify the results of your calculations.
Exploring Quadratic Equations
Quadratic equations are mathematical expressions that can be written in the standard form \( ax^2 + bx + c = 0 \). They are called 'quadratic' because the highest degree of the variable \( x \) is 2. Each quadratic equation is characterized by three coefficients:
  • \( a \): The leading coefficient, which is associated with \( x^2 \).
  • \( b \): The middle coefficient, linked to the linear term \( x \).
  • \( c \): The constant term or free number, which does not involve \( x \).
The primary goal when working with a quadratic equation is to find the values of \( x \) that make the equation true, known as the solutions or roots of the equation. In our problem, the equation \( x^{2}-5x+3=0 \) has coefficients \( a=1 \), \( b=-5 \), and \( c=3 \). These will be used to apply various methods for solving, one of which is the quadratic formula.
Solving Equations Using the Quadratic Formula
The quadratic formula is a reliable method for solving quadratic equations when factoring is difficult or impractical. The formula is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]To solve a quadratic equation using this formula, follow these steps:
  • Identify the coefficients \( a \), \( b \), and \( c \) from the equation.
  • Calculate the discriminant \( b^2 - 4ac \).
  • Substitute \( a \), \( b \), and \( c \) into the quadratic formula.
  • Simplify under the square root and solve for \( x \).
In our example with \( x^{2}-5x+3=0 \):
  • Coefficients are \( a=1 \), \( b=-5 \), \( c=3 \).
  • Discriminant is \( 13 \), leading to real solutions.
  • Use the formula to find roots: \( x = \frac{5 \pm \sqrt{13}}{2} \).
You'll get two solutions: \( x = \frac{5+\sqrt{13}}{2} \) and \( x = \frac{5-\sqrt{13}}{2} \). The quadratic formula is a versatile and powerful tool that always yields the solutions whenever they exist.

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Most popular questions from this chapter

In Exercises \(97-100,\) let \(f(t)=-t^{2}\) and \(g(x)=x^{2}-1\). Evaluate \((f \circ f)(-1)\)

The point (2,4) on the graph of \(f(x)=x^{2}\) has been shifted horizontally to the point \((-3,4) .\) Identify the shift and write a new function \(g(x)\) in terms of \(f(x)\).

A security firm currently has 5000 customers and charges \(\$ 20\) per month to monitor each customer's home for intruders. A marketing survey indicates that for each dollar the monthly fee is decreased, the firm will pick up an additional 500 customers. Let \(R(x)\) represent the revenue generated by the security firm when the monthly charge is \(x\) dollars. Find the value of \(x\) that results in the maximum monthly revenue.

A chartered bus company has the following price structure. A single bus ticket costs \(\$ 30 .\) For each additional ticket sold to a group of travelers, the price per ticket is reduced by \(\$ 0.50 .\) The reduced price applies to all the tickets sold to the group. (a) Calculate the total cost for one, two, and five tickets. (b) Using your calculations in part (a) as a guide, find a quadratic function that gives the total cost of the tickets. (c) How many tickets must be sold to maximize the revenue for the bus company?

The production cost, in dollars, for \(x\) color brochures is \(C(x)=500+3 x .\) The fixed cost is \(\$ 500\) since that is the amount of money needed to start production even if no brochures are printed. (a) If the fixed cost is decreased by \(\$ 50,\) find the new cost function. (b) Graph both cost functions and interpret the effect of the decreased fixed cost.
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