How to Solve Quadratic Equations Without Formulas or Calculators

When faced with a quadratic equation like 2x2 - 5x - 3 0, it's tempting to immediately resort to formulas or calculators. However, there are alternative methods—one being factoring, and the other being completing the square. Let's explore these techniques and understand how they can be applied.

Factoring

Factoring a quadratic equation involves breaking down the equation into factors that multiply to give the original equation. For the equation 2x2 - 5x - 3 0, the goal is to express it in the form 2x - a(x - b) where x a · b are the roots of the equation.

First, identify two numbers that multiply to 2 × -3 -6 and add to -5. In this case, the numbers are -6 and 1. We can then split the middle term -5x into -6x x.

2x2 - 6x x - 3 0

Rearrange and factor by grouping:

2x(x - 3) 1(x - 3) 0

(2x 1)(x - 3) 0

Solving for x, we get:

x -1/2 or x 3

Completing the Square

Completing the square is an alternative method that transforms the equation into a perfect square trinomial. This method is particularly useful when factoring is not straightforward. Let's demonstrate this method using the same equation:

2x2 - 5x - 3 0

First, divide the entire equation by the coefficient of x2 to get a leading coefficient of 1:

x2 - (5/2)x - (3/2) 0

Next, move the constant term to the other side:

x2 - (5/2)x 3/2

Now, add and subtract the square of half the coefficient of x on the left side:

x2 - (5/2)x (5/4)2 - (5/4)2 3/2

x2 - (5/2)x (5/4)2 3/2 (5/4)2

(x - 5/4)2 3/2 25/16

(x - 5/4)2 24/16 25/16 49/16

(x - 5/4)2 (7/4)2

Take the square root of both sides:

x - 5/4 ±7/4

Solve for x by adding 5/4 to both sides:

x 5/4 ± 7/4

This gives us two roots:

x 3 or x -1/2

Vietas Formulas

Adobe another approach, we can use Vietas formulas. These are relationships between the coefficients of a polynomial and its roots. For a quadratic equation nx2 - Sx P 0, where S is the sum of the roots and P is the product of the roots. For example, if we have:

x2 - 7x 12 0

Then S 7 and P 12. We can find two numbers that add up to 7 and multiply to give 12. In this case, those numbers are 3 and 4.

Conclusion

Without relying on formulas or calculators, you can find solutions to quadratic equations using factoring, completing the square, and leveraging Vietas formulas. These methods provide a deeper understanding of the underlying concepts and are invaluable for mathematical problem-solving. No matter the method, the key is to break down the problem step-by-step.

Keywords

The keywords for this article include: Quadratic Equations, Factoring, Completing the Square.