Question
upstudy study bank question image url

Solve the Quadratic Equation
From Fist Pinciple.
to

Ask by Gibbs Hill. in Australia
Dec 27,2024

Upstudy AI Solution

Tutor-Verified Answer

Answer

The quadratic formula for the equation is:
This formula provides the solutions for in the quadratic equation.

Solution

Certainly! Let’s derive the Quadratic Formula from the first principles for the general quadratic equation:
where .

Step 1: Start with the Standard Form

Begin with the standard quadratic equation:

Step 2: Divide Both Sides by

To simplify, divide every term by (since ):

Step 3: Move the Constant Term to the Right Side

Step 4: Complete the Square

To complete the square, take half of the coefficient of , square it, and add it to both sides.
  1. Find half of :
  1. Square it:
  1. Add this square to both sides:

Step 5: Rewrite the Left Side as a Perfect Square

The left side now forms a perfect square:

Step 6: Simplify the Right Side

Combine the terms on the right side:

Step 7: Take the Square Root of Both Sides

Step 8: Solve for

Subtract from both sides:
Combine the terms over a common denominator:

Quadratic Formula

Thus, the solutions to the quadratic equation are given by:

Addressing the Provided Expression

You mentioned the expression:
It seems there might be a misunderstanding in the formula you’ve provided. Let’s clarify the correct components:
  1. Discriminant: The term under the square root in the quadratic formula is , not or .
  2. Denominator: The denominator in the quadratic formula is , not 29.
  3. Overall Structure: The correct quadratic formula has the square root applied solely to the discriminant.

Correct Formula Comparison

Ensure that:
  • The discriminant is .
  • The entire numerator (including the square root term) is over .
If you have specific values for , , and , feel free to share them, and I can help compute the solutions using the correct quadratic formula!

Answered by UpStudy AI and reviewed by a Professional Tutor

error msg
Explain
Simplify this solution

The Deep Dive

Let’s break this down! A quadratic equation of the form can be graphed as a parabola. The solutions, or roots, can be derived using the quadratic formula . Notably, the you mentioned seems a bit off; it’s actually under the square root for the correct formula.
Now, to bring some excitement into quadratic equations, consider this: they pop up everywhere, from physics scenarios like projectile motion to finance when calculating profit maximization! You could say they’re the unsung heroes of both math and the real world – bringing order where chaos might reign!

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