/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 13 Solve the equation. Tell which m... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 13

Solve the equation. Tell which method you used. \(z^{2}+11 z+30=0\)

Short Answer

Expert verified
The solutions to the equation are \(z = 5\) and \(z = 6\). The method used was the Quadratic Formula.

Step by step solution

01

Identifying coefficients

The quadratic equation is \(z^{2}+11 z+30=0\). Here, the coefficients are \(a=1\), \(b=11\), and \(c=30\).
02

Calculating the discriminant

The discriminant is calculated by \(b^{2}-4ac\). So, substituting the values, we have \(11^2 - 4*1*30 = 121 - 120 = 1\). As this is a positive number we can expect two distinct real roots for this quadratic equation.
03

Applying the Quadratic formula

Now we apply these values to the quadratic formula. We find that \(z = \frac{-(-11)\pm\sqrt{1}}{2*1}\), which simplifies to \(z = \frac{11\pm 1}{2}\). This results in two solutions: \(z = \frac{11+1}{2} = 6\) and \(z = \frac{11-1}{2} = 5\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Discriminant
The discriminant is an important part of solving quadratic equations. It helps us understand what kind of solutions we can expect. In a quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant \(D\) is given by the formula \(D = b^2 - 4ac\).

The value of the discriminant can tell us a lot about the roots of the equation:
  • If \(D > 0\), there are two distinct real roots. That means the parabola intersects the x-axis at two points.
  • If \(D = 0\), there is exactly one real root, or in other words, the parabola touches the x-axis at one point (vertex).
  • If \(D < 0\), there are no real roots, and the solutions are complex or imaginary numbers because the parabola does not intersect the x-axis.
For the equation \(z^2 + 11z + 30 = 0\), we calculated the discriminant as 1, which is positive, indicating two distinct real roots.
Quadratic Formula
The quadratic formula is a powerful tool for finding the roots of any quadratic equation \(ax^2 + bx + c = 0\). Here, the formula is:
  • \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
This formula allows for directly calculating the roots without needing factors of the equation. Here鈥檚 how it works:
  • First, ensure that all components \(a\), \(b\), and \(c\) from the equation are identified correctly.
  • Use these values in the formula to solve for \(x\), taking note of the plus-minus symbol \(\pm\), which results in the two possible solutions.
For our example \(z^2 + 11z + 30 = 0\), we identified \(a = 1\), \(b = 11\), and \(c = 30\). Substituting these into the formula gives us the roots \(z = 6\) and \(z = 5\). This method is reliable and works irrespective of whether the quadratic is factorable.
Factorization Method
Factorization is often the simplest way to solve a quadratic if the equation can easily be broken down into two factors. In equations like \(x^2 + bx + c = 0\), we aim to express them in a form that resembles (x + p)(x + q) = 0, where both p and q are numbers.

Here's how you factorize:
  • First, seek two numbers that multiply to \(c\) and add up to \(b\).
  • Once these numbers are found, rewrite the quadratic expression as a product of two binomials.
  • Solve each binomial equation for zero to find the roots of the equation.
In the given equation \(z^2 + 11z + 30 = 0\), the factors of 30 that add up to 11 are 5 and 6. Hence, the equation can be factorized as \((z + 5)(z + 6) = 0\). Setting each factor to zero gives the solutions \(z = -5\) and \(z = -6\). Factorization offers an intuitive understanding of the solutions by showing how the quadratic is constructed.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Which of the following is the solution of the equation \(-4 x^{2}+24 x-36=0 ?\) A) -6 B) -3 C) 2 D) 3

In Exercises \(69-72,\) you are tutoring a friend and want to create some quadratic equations that can be solved by factoring. Find a quadratic equation that has the given solutions and explain the procedure you used to obtain the equation. $$4 and -3$$

Factor the expression. Tell which special product factoring pattern you used. $$4 b^{2}-40 b+100$$

Factor the expression. Tell which special product factoring pattern you used. $$-2 x^{2}+52 x-338$$

Simplify the expression. $$\frac{-6 \sqrt{12}}{\sqrt{4}}$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.