Prove that ax² + bx + c = -b ± √(b² - 4ac) / 2a

Steps to Solve

1. Start with the standard quadratic equation

We begin with the quadratic equation in the standard form:

$$ ax^2 + bx + c = 0 $$

2. Divide by 'a'

To simplify the equation, divide every term by $a$:

$$ x^2 + \frac{b}{a}x + \frac{c}{a} = 0 $$

3. Rearrange the equation

Next, rearrange the equation to isolate the constant term:

$$ x^2 + \frac{b}{a}x = -\frac{c}{a} $$

4. Complete the square

To complete the square on the left side, add $\left(\frac{b}{2a}\right)^2$ to both sides:

$$ x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2 $$

This simplifies to:

$$ \left(x + \frac{b}{2a}\right)^2 = -\frac{c}{a} + \frac{b^2}{4a^2} $$

5. Combine terms

On the right side, find a common denominator and combine:

$$ \left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2} $$

6. Take the square root of both sides

Now, take the square root of both sides:

$$ x + \frac{b}{2a} = \pm \sqrt{\frac{b^2 - 4ac}{4a^2}} $$

7. Solve for x

This can be rewritten as:

$$ x + \frac{b}{2a} = \frac{\pm \sqrt{b^2 - 4ac}}{2a} $$

Finally, isolate $x$:

$$ x = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a} $$

So,

$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$