x² + bx + c = 0

mathematics question and need the explanation and answer to help me learn.

powerful tool for solving quadratic equations. By understanding the completing the square method and memorizing the formula, you will be well-prepared to tackle any quadratic equation.
Let’s derive the quadratic formula using the method of completing the square. We start with the equation:
x² + bx + c = 0
To complete the square, we take the coefficient of x, b, divide it by 2, and square it. This gives us (b/2)² = (b²/4).
Now, we add and subtract (b²/4) to the equation:
x² + bx + (b²/4) – (b²/4) + c = 0
Next, we group the first three terms and rewrite the equation:
(x² + bx + (b²/4)) – (b²/4) + c = 0
Now, we can rewrite the first three terms as a perfect square:
(x + b/2)² – (b²/4) + c = 0
Simplifying further, we have:
(x + b/2)² – (b²/4) + c – (b²/4) = 0
Combining the constants, we get:
(x + b/2)² – (b²/4) + c – (b²/4) = 0
(x + b/2)² – (b²/4) + (4c – b²)/4 = 0
Now, we can write the equation in the form:
(x + b/2)² = (b² – 4ac)/4
Taking the square root of both sides, we have:
x + b/2 = ±√(b² – 4ac)/2
Finally, rearranging the equation, we get the quadratic formula:
x = (-b ± √(b² – 4ac))/2a
That’s it! We have derived the quadratic formula by completing the square. This method is simple and easy to understand. Remember, completing the square involves comparing the given equation to the standard form x² + bx + c = 0 and manipulating it until we get a perfect square. While there may be other methods to derive the quadratic formula, this approach is beginner-friendly.
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