Deriving the Quadratic Formula

The quadratic equation, written in the general form as ax² + bx + c = 0 is derived using the steps involved in completing the square.

Steps to Derive the Quadratic Formula

To derive the quadratic formula, complete the square and solve for the variable x.

Step 1: In a quadratic function y = ax² + bx + c, where a, b, and c are real numbers and a ≠ 0, let y = 0, such that:

ax² + bx + c = 0

Step 2: Move ‘c’ to the right of the equation by subtracting both sides by ‘c’

ax² + bx + c – c = 0 – c

=> ax² + bx = – c

Step 3: Divide both sides of the equation by ‘a’

=> ${\dfrac{1}{a}( ax^{2}+bx= -c)}$

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

Step 4: Identify the coefficient of x, divide by 2, square the total term and simplify

=> ${\left( \dfrac{\dfrac{b}{a}}{2}\right) ^{2}=\left( \dfrac{b}{2a}\right) ^{2}}$

=> ${\dfrac{b^{2}}{4a^{2}}}$

Step 5: Add the result of step 5 to both sides of the equation

=> ${x^{2}+\dfrac{b}{a}x+\dfrac{b^{2}}{4a^{2}}=-\dfrac{c}{a}+\dfrac{b^{2}}{4a^{2}}}$

Step 6: Simplify the right side of the equation

=> ${x^{2}+\dfrac{b}{a}x+\dfrac{b^{2}}{4a^{2}}=\dfrac{-c\times 4a}{a\times 4a}+\dfrac{b^{2}}{4a^{2}}}$

=> ${x^{2}+\dfrac{b}{a}x+\dfrac{b^{2}}{4a^{2}}=${\dfrac{-4ac}{4a^{2}}+\dfrac{b^{2}}{4a^{2}}}$

=> ${x^{2}+\dfrac{b}{a}x+\dfrac{b^{2}}{4a^{2}}= ${\dfrac{b^{2}-4ac}{a^{2}}}$

Step 7: Express the trinomial as a perfect square

=> ${[ +\dfrac{b}{2a}) ^{2}=b^{2}\dfrac{-4ac}{4a^{2}}}$

Step 8: Square root both sides to remove the exponent 2

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

Step 9: Simplify and add ≠ on the right side of the equation

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

Step 10: Subtract both sides by ${\dfrac{b}{2a}}$ to keep only ‘x’ on the left side and simplify

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

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

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

This gives us the quadratic formula we know and use for our calculations.