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.
