4x ^ 2 – 5x – 12 = 0 : How to Solve Quadratic Equations

Jessy

1 year ago

Quadratic equations are equations of the form

Ax2+Bx+C=0, where B and C are constants. Quadratic equations can be used to model various phenomena, such as projectile motion, area, optimization, etc. In this blog post, we will learn how to solve quadratic equations using three methods: factoring, completing the square, and the quadratic formula. We will use the example to illustrate each method.

Factoring

Factoring is a method of solving quadratic equations by finding two factors of the equation that multiply to give zero. To factor a quadratic equation, we need to find two numbers that add up to $B$ and multiply to. For example, for the equation, $4 x^{2} - 5 x - 12 = 0$ We need to find two numbers that add up to $- 5$ and multiply to $- 48$ (the product of. However, in this case, there are no such numbers, so we cannot factor the equation. Therefore, factoring is not a suitable method for this equation.

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Completing the Square

Completing the square is a method of solving quadratic equations by transforming the equation into a perfect square form. To complete the square, we need to follow these steps:

Divide both sides of the equation by $A to make the coefficient of x^{2} equal to one.$
Add or subtract a constant term to both sides of the equation to make the coefficient of $x$ a perfect square.
Write the left side of the equation as a binomial squared.
Take the square root of both sides of the equation and solve for x.

For example, for the equation

4 x^{2} - 5 x - 12 = 0

We can complete the square as follows:

Divide both sides by $4 : x^{2} - 45 x - 3 = 0$
Add $6425$
to both sides:
$x^{2} - 45 x + 6425 = 64217$
Write the left side as a binomial squared: $(x - 85)^{2} = 64217$
Take the square root of both sides. $x - 85 = \pm 64217$
Solve for $x : x = 85 \pm 64217$

Quadratic Formula

The quadratic formula is a general formula that can be used to solve any quadratic equation. The quadratic formula is derived from completing the square and states that:

x = 2 A - B \pm B ^{2} - 4 A C

To use the quadratic formula, we need to identify the values of $A$ , $B$ , and $C$ from the equation and plug them into the formula. For example, for the equation

4 x^{2} - 5 x - 12 = 0

We have:

$A = 4$
$B = - 5$
$C = - 12$

Plugging these values into the formula, we get:

x = 2 ( 4 ) - ( -5 ) \pm ( -5 ) ^{2} - 4 ( 4 ) ( -12 )

Simplifying, we get:

x = 85 \pm 217

Conclusion

Quadratic equations are common in mathematics and can be solved using different methods. In this blog post, we learned how to solve quadratic equations using factoring, completing the square, and the quadratic formula. We used the example of $4 x^{2} - 5 x - 12 = 0$ to demonstrate each method. We found that factoring was not possible for this equation, while completing the square and the quadratic formula gave us the same solutions: $x = 85 \pm 64217$ We hope this blog post was helpful and informative for you. If you have any questions or feedback, please leave a comment below. Thank you for reading!