/*! 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 Write in standard form. Use the ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 13

Write in standard form. Use the quadratic formula to solve the equation. $$-14 x=-2 x^{2}+36$$

Short Answer

Expert verified
The solutions of the quadratic equation are \(x = 2\) and \(x = -9\)

Step by step solution

01

Rearrange the Equation in Standard Form

Rearrange \( -14x = -2x^{2} + 36 \) to the standard form by adding \(2x^{2}\) and \(14x\) to both sides. The equation now reads: \(2x^{2} + 14x - 36 = 0\)
02

Identify a, b and c

From the rearranged equation, \(a = 2\), \(b = 14\), and \(c = -36\)
03

Substitute values into the Quadratic Formula

Substitute the values into the quadratic formula: \[x= {-14卤\sqrt{14^{2}-4*2*(-36)} \over 2*2}\] This simplifies to: \[x= {-14卤\sqrt{196+288} \over 4}\] Further simplifying gives: \[x= {-14卤\sqrt{484} \over 4}\], which, when simplified yet again, gives: \[x= {-14卤22 \over 4}\] This subsequently gives two roots: \(x1 = 2\) and \(x2 = -9\).

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.

Standard Form
When working with quadratic equations, the standard form is an essential structure. Standard form is expressed as \( ax^2 + bx + c = 0 \). This format makes it easier to identify the coefficients needed when using the quadratic formula.
To convert any equation into standard form:
  • Make sure the quadratic term \( ax^2 \) is on one side of the equation.
  • Rearrange other terms to follow suit: the linear term \( bx \) and the constant \( c \) should be on the same side.
  • Ensure the right side equals zero.
In our original equation, \(-14x = -2x^2 + 36\), we can rearrange it to get: \(2x^2 + 14x - 36 = 0\). This gives us a clear pathway to correctly applying the quadratic formula.
Quadratic Equation
A quadratic equation is an algebraic expression of the second degree. It forms a curve known as a parabola when graphed. The general form of a quadratic equation is \( ax^2 + bx + c = 0 \), where:
  • \(a\) is the coefficient of \(x^2\)
  • \(b\) is the coefficient of \(x\)
  • \(c\) is the constant term
In our problem, identifying \(a\), \(b\), and \(c\) after rearranging to standard form is crucial. We have \(a = 2\), \(b = 14\), and \(c = -36\). These coefficients are vital for solving the quadratic equation using methods like factorization, completing the square, or the quadratic formula.
Roots of Quadratic Equations
The roots of a quadratic equation are the solutions where the equation equals zero. These roots are where the parabola intercepts the x-axis on a graph.
Using the quadratic formula, \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \), you can determine the roots with ease. Make sure the equation is in standard form to appropriately substitute your \(a\), \(b\), and \(c\) values.
From our equation \(2x^2 + 14x - 36 = 0\), substituting into the formula gives:
  • First, calculate the discriminant: \(b^2 - 4ac = 196 + 288 = 484\)
  • Square root the discriminant: \(\sqrt{484} = 22\)
  • Use the formula to find \(x_1 = \frac{{-14 + 22}}{4} = 2\) and \(x_2 = \frac{{-14 - 22}}{4} = -9\)
Thus, the roots are \(x = 2\) and \(x = -9\), giving you the points where the parabola cuts through the x-axis.

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

Sketch the graph of the function. Label the vertex. y=-5 x^{2}-0.5 x+0.5

Evaluate \(\sqrt{b^{2}-4 a c}\) for the given values. $$a=-2, b=8, c=-8$$

Use a calculator to evaluate the expression. Round the results to the nearest hundredth. $$\frac{1 \pm 6 \sqrt{8}}{6}$$

LOGICAL REASONING Consider the equation \(a x^{2}+b x+c=0\) and use the quadratic formula to justify the statement. If \(b^{2}-4 a c\) is positive, then the equation has two solutions.

Use a graph to solve the linear system. Check your solution algebraically. (Review 7.1 ) $$\begin{aligned}&4 x+5 y=20\\\&\frac{5}{4} x+y=4\end{aligned}$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Calculus

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.