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The “horrible looking”

below

is actually derived using the steps involved in completing the square. It stems from the fact that any quadratic function or equation of the form $y = a{x^2} + bx + c$ can be solved for its roots. The “roots” of the quadratic equation are the points at which the graph of a quadratic function (the graph is called the parabola) hits, crosses or touches the x-axis known as the $x$-intercepts.

So to find the roots or x-intercepts of $y = a{x^2} + bx + c$, we need to let $y = 0$. That means we have

ax2 + bx + c = 0

From here, I am going to apply the usual steps involved in

completing the square

to arrive at the quadratic formula.

## Steps on How to Derive the Quadratic Formula

Derivation of the quadratic formula is easy! Here we go.

• Step 1: Let $y = 0$ in the general form of the quadratic function $y = a{x^2} + bx + c$ where $a$, $b$, and $c$ are real numbers but $a ne 0$.
• Step 2: Move the constant $color{red}c$ to the right side of the equation by subtracting both sides by $color{red}c$ .
• Step 3: Divide the entire equation by the coefficient of the squared term which is $large{a}$.
• Step 4: Now identify the coefficient of the linear term $large{x}$.
• Step 5: Divide it by $2$ and raise it to the 2nd power. Then simplify it further.
• Step 6: Add the output of step #5 to both sides of the equation.
• Step 7: Simplify the right side of the equation. Be careful when you

. Make sure that you find the correct

Least Common Denominator

• Step 8: Express the trinomial on the left side of the equation as the square of a binomial.
• Step 9: Take the square root of both sides of the equation to eliminate the exponent $2$ of the binomial.
• Step 10: Simplify. Make sure that you attach the $color{red} pm$ on the right side of the equation. The left side no longer contains the power $2$.
• Step 11: Keep the variable $x$ on the left side by subtracting both sides by $Large{b over {2a}}$.
• Step 12: Simplify and we are done!

I hope that you find the step-by-step solution helpful in figuring out how the quadratic formula is derived using the method of completing the square.

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