Question
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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.
- Find half of
:
- Square it:
- 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:
-
Discriminant: The term under the square root in the quadratic formula is
, not or . -
Denominator: The denominator in the quadratic formula is
, not 29. -
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
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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!