Login
Premium
Home Question Bank Math Algebra
Question
upstudy study bank question image url

\( x = \frac { - b \pm \sqrt { b ^ { 2 } - 4 a c } } { 2 a } \)

Ask by Parsons Watkins. in Argentina
Mar 11,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

La fórmula cuadrática es: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Esta fórmula se usa para encontrar las soluciones de una ecuación cuadrática \( ax^2 + bx + c = 0 \).

Solution

Sign in to Unlock Answers for Free!

A Learning Platform Trusted by Millions of Real Students and Teachers.

star-icon Unlock

Answered by UpStudy AI and reviewed by a Professional Tutor

The Deep Dive

The quadratic formula, which you’ve shared, has a rich historical background! It dates back to ancient civilizations, with roots in Babylonian mathematics around 2000 BC, where problems resembling quadratic equations were solved using geometric methods. The formula itself was later formalized in the Islamic Golden Age by mathematicians like Al-Khwarizmi, who laid the groundwork for algebra as we know it today. In modern times, this formula finds real-world applications in fields such as physics, engineering, and finance, helping to model trajectories, optimize project budgets, and assess risks, respectively. For example, when analyzing projectile motion, the quadratic formula can determine the time at which an object reaches a certain height, marrying math with physical phenomena in our daily lives!

Related Questions

himpunan penyelesaian dari sistem persamaan \( 2 x+2 y=4 \) dan \( 3 x+y=6 \) adalah
Algebra Indonesia Mar 17, 2025
Question 6 (1 point) state the total number of complex roots, the possible number of real and imaginary roots and the possible rational roots for each equation. Then find all roots. \[ x^{3}-2 x^{2}-3 x+6=0 \] a \# of complex roots: 6 Possible \# of real roots: 3 or 1 Possible \# of imaginary roots: 2 or 0 Possible rational roots: \( \pm 1, \pm 2 \pm 3, \pm 6 \) Roots: \( -1, \pm \sqrt{3} \) b \# of complex roots: 6 Possible \# of real roots: 3 or 1 Possible \# of imaginary roots: 2 or 0 Possible rational roots: \( \pm 1, \pm 2 \pm 3, \pm 6 \) Roots: \( 2, \pm \sqrt{3} \) c \# of complex roots: 3 Possible \# of real roots: 3 or 1 Possible \# of imaginary roots: 2 or 0 Possible rational roots: \( \pm 1, \pm 2 \pm 3, \pm 6 \) Roots \( 2, \pm \sqrt{3} \) d \# of complex roots: 3 Possible \# of real roots: 3 or 1 Possible \# of imaginary roots: 2 or 0 Possible rational roots: \( \pm 1, \pm 2 \pm 3, \pm 6 \) Roots: \( -1, \pm \sqrt{3} \)
Algebra United States Mar 17, 2025
Summarize all pertinent information obtained by applying the graphing strategy and sketch the graph of \( y=f(x) \). \[ f(x)=\frac{x^{2}+13 x+36}{x^{2}+8 x+16} \] tina the aomain or \( \mathrm{r}(\mathrm{X}) \). seiect the correct cnoice deiow ana, it necessary, till in the answer dox to complete your cnoice. A. The domain is all real \( x \), except \( x= \) \( \square \) (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. The domain is all real \( x \). Find the \( x \)-intercepts of \( f(x) \). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The \( x \)-intercept(s) is/are at \( x= \) \( \square \) . (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. There are no \( x \)-intercepts. Find the \( y \)-intercepts of \( f(x) \). Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
Algebra United States Mar 17, 2025
State the total number of complex roots, the possible number of real and imaginary roots and the possible rational roots for each equation. Then find all roots. \[ x^{3}-1=0 \] a \# of complex roots: 3 Possible \# of real roots: 3 or 1 Possible \# of imaginary roots: 2 or 0 Possible rational roots: \( \pm 1 \) \[ \text { Roots: } \pm 1, \frac{-1+i \sqrt{3}}{2} \] b \[ 1, \frac{-1+i \sqrt{3}}{2}, \frac{-1-i \sqrt{3}}{2} \text { \# of complex roots: } 3 \] Possible \# of real roots: 3 or 1 Possible \# of imaginary roots: 2 or 0 Possible rational roots: \( \pm 1 \) Roots: c \# of complex roots: 3 Possible \# of real roots: 3 or 1 Possible \# of imaginary roots: 2 or 0 Possible rational roots: \( \pm 1 \) \[ \text { Roots: }-1, \frac{-1+i \sqrt{2}}{5}, \frac{-1-i \sqrt{2}}{5} \] d \# of complex roots: 3 Possible \# of real roots: 3 or 1 Possible \# of imaginary roots: 2 or 0 Possible rational roots: \( \pm 1 \) Roots: \[ -1, \frac{i \sqrt{3}}{2}, \frac{-i \sqrt{3}}{2} \]
Algebra United States Mar 17, 2025
Question 7 (1 point) State the total number of complex roots, the possible number of real and imaginary roots and the possible rational roots for each equation. Then finc all roots \[ x^{6}-x^{4}-x^{2}+1=0 \] a \# of complex roots: 6 Possible \# of real roots: \( 6,4,2 \), or 0 Possible \# of imaginary roots: \( 6,4,2 \), or 0 Possible rational roots: \( \pm 1 \) Roots: \( \{-1, i,-i\} \) b \# of complex roots: 6 Possible \# of real roots: \( 6,4,2 \), or 0 Possible \# of imaginary roots: \( 6,4,2 \), or 0 Possible rational roots: \( \pm 1 \) Roots: \( \{1 \) mult. \( 2,-1 \) mult. \( 2, i,-i\} \) c \# of complex roots: 6 Possible \# of real roots: \( 6,4,2 \), or 0 Possible \# of imaginary roots: \( 6,4,2 \), or 0 Possible rational roots: \( \pm 1 \) Roots: \( \{1 \) mult. 2, -1 mult. 2\} d \# of complex roots: 6 Possible \# of real roots: \( 6,4,2 \), or 0 Possible \# of imaginary roots: \( 6,4,2 \), or 0 Possible rational roots: \( \pm 1 \) Roots: \( \{1,-1,-2,2, i,-i\} \) Review Answers
Algebra United States Mar 17, 2025

Latest Algebra Questions

3. Write out Cayley tables for groups formed by the symmetries of a rectangle and for \( \left(\mathbb{Z}_{4 y},+\right) \). How many elements are in each group? Are the groups the same? Why or why not?
Algebra United States Mar 17, 2025
himpunan penyelesaian dari sistem persamaan \( 2 x+2 y=4 \) dan \( 3 x+y=6 \) adalah
Algebra Indonesia Mar 17, 2025
Question 4 (1 point) State the total number of complex roots, the possible number of real and imaginary roots and the possible rational roots for each equation. Then find all roots. \( \begin{array}{l}x^{3}+3 x^{2}-x-3=0 \\ \text { a of complex roots: } 3 \\ \text { Possible \# of real roots: } 3 \text { or } 1 \\ \text { Possible \# of imaginary roots: } 2 \text { or } 0 \\ \text { Possible rational roots: } \pm 1, \pm 3 \\ \text { Roots: }\{-3,1,-1\} \\ \text { \# of complex roots: } 3 \\ \text { Possible \# of real roots: } 3 \text { or } 1 \\ \text { Possible \# of imaginary roots: } 2 \text { or } 0 \\ \text { Possible rational roots: } \pm 1, \pm 3 \\ \text { Roots: }\{3,1,-1\} \\ \text { \# of complex roots: } 3 \\ \text { Possible \# of real roots: } 3 \text { or } 1 \\ \text { Possible \# of imaginary roots: } 2 \text { or } 0 \\ \text { Possible rational roots: } \pm 1, \pm 3 \\ \text { Roots: }\{-3,-1,-1\} \\ \text { \# of complex roots: } 3 \\ \text { Possible \# of real roots: } 3 \text { or } 1 \\ \text { Possible \# of imaginary roots: } 2 \text { or } 0 \\ \text { Possible rational roots: } \pm 1, \pm 3 \\ \text { Roots: }\{3, \text { double root at } 1\}\end{array} \) \( \begin{array}{l}\text { d }\end{array} \)
Algebra United States Mar 17, 2025
Summarize all pertinent information obtained by applying the graphing strategy and sketch the graph of \( y=f(x) \). \[ f(x)=\frac{x^{2}+13 x+36}{x^{2}+8 x+16} \] tina the aomain or \( \mathrm{r}(\mathrm{X}) \). seiect the correct cnoice deiow ana, it necessary, till in the answer dox to complete your cnoice. A. The domain is all real \( x \), except \( x= \) \( \square \) (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. The domain is all real \( x \). Find the \( x \)-intercepts of \( f(x) \). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The \( x \)-intercept(s) is/are at \( x= \) \( \square \) . (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. There are no \( x \)-intercepts. Find the \( y \)-intercepts of \( f(x) \). Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
Algebra United States Mar 17, 2025
State the total number of complex roots, the possible number of real and imaginary roots and the possible rational roots for each equation. Then find all roots. \[ x^{3}-1=0 \] a \# of complex roots: 3 Possible \# of real roots: 3 or 1 Possible \# of imaginary roots: 2 or 0 Possible rational roots: \( \pm 1 \) \[ \text { Roots: } \pm 1, \frac{-1+i \sqrt{3}}{2} \] b \[ 1, \frac{-1+i \sqrt{3}}{2}, \frac{-1-i \sqrt{3}}{2} \text { \# of complex roots: } 3 \] Possible \# of real roots: 3 or 1 Possible \# of imaginary roots: 2 or 0 Possible rational roots: \( \pm 1 \) Roots: c \# of complex roots: 3 Possible \# of real roots: 3 or 1 Possible \# of imaginary roots: 2 or 0 Possible rational roots: \( \pm 1 \) \[ \text { Roots: }-1, \frac{-1+i \sqrt{2}}{5}, \frac{-1-i \sqrt{2}}{5} \] d \# of complex roots: 3 Possible \# of real roots: 3 or 1 Possible \# of imaginary roots: 2 or 0 Possible rational roots: \( \pm 1 \) Roots: \[ -1, \frac{i \sqrt{3}}{2}, \frac{-i \sqrt{3}}{2} \]
Algebra United States Mar 17, 2025
Prev Next
Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy