The Standard Form of the Quadratic Equation and Its Solution

Understanding how to convert and solve a quadratic equation in its standard form is a fundamental skill in algebra. This article will help you navigate the process using the example 3xx4 7.

Converting the Equation into Standard Form

The given equation is 3xx4 7. Assuming the 'xx' symbol is a typo and represents multiplication, let's convert it into standard form. The standard form of a quadratic equation is generally expressed as ax^2 bx c 0. Here's how to do it:

3xx 4 3x^2 - 12x

3x^2 - 12x 7

3x^2 - 12x - 7 0

Now, the equation is in the standard form 3x^2 - 12x - 7 0.

Solving the Quadratic Equation

There are several methods to solve a quadratic equation in standard form. The most common are the Quadratic Formula and Completing the Square. Let's explore both methods with our example:

Quadratic Formula

The Quadratic Formula is a powerful tool for solving any quadratic equation. It is expressed as:

x frac{-b pm sqrt{b^2 - 4ac}}{2a}

For the equation 3x^2 - 12x - 7 0, the coefficients are a 3, b -12, and c -7. Plugging these values into the Quadratic Formula, we get:

x frac{-(-12) pm sqrt{(-12)^2 - 4(3)(-7)}}{2(3)}

x frac{12 pm sqrt{144 84}}{6}

x frac{12 pm sqrt{228}}{6}

x frac{12 pm 2sqrt{57}}{6}

x -2 pm frac{sqrt{57}}{3}

Therefore, the solutions are:

x_1 -2 frac{sqrt{57}}{3}

x_2 -2 - frac{sqrt{57}}{3}

Completing the Square

Another method to solve the equation is by completing the square. Here's how it is done:

Start with the standard form: 3x^2 - 12x - 7 0

Divide the entire equation by 3 to simplify:

x^2 - 4x - frac{7}{3} 0

Move the constant term to the right side:

x^2 - 4x frac{7}{3}

To complete the square, add and subtract the square of half the coefficient of (x):

x^2 - 4x 4 frac{7}{3} 4

This can be simplified to:

(x - 2)^2 frac{7}{3} frac{12}{3}

(x - 2)^2 frac{19}{3}

Taking the square root of both sides:

x - 2 pm sqrt{frac{19}{3}}

Therefore:

x 2 pm sqrt{frac{19}{3}}

Again, the solutions are:

x_1 -2 sqrt{frac{19}{3}}

x_2 -2 - sqrt{frac{19}{3}}

Graphical Interpretation

The solutions obtained are the x-intercepts of the parabola represented by the equation (y 3x^2 - 12x - 7). The parabola intersects the x-axis where (y 0). The graph of this parabola would look as follows:

Key Takeaways

Quadratic Equation: An equation of the form (ax^2 bx c 0) where (a, b, c) are constants and (a eq 0).

Standard Form: Refers to the form (ax^2 bx c 0) which is essential for solving quadratic equations.

Quadratic Formula: A method for solving quadratic equations of the form (ax^2 bx c 0), given by (x frac{-b pm sqrt{b^2 - 4ac}}{2a}).