How to Solve the Quadratic Equation 3x^2 - 23x - 110 Using Different Methods

Quadratic equations are a fundamental part of algebra and often require multiple methods to find the roots. The equation 3x^2 - 23x - 110 0 is a classic example of a quadratic equation. In this article, we will explore different methods to solve this equation step-by-step. We will explore the quadratic formula, factoring, and completing the square.

Solution Using the Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form (ax^2 bx c 0). For the equation (3x^2 - 23x - 110 0), the coefficients are:

(a 3) (b -23) (c -110)

To solve this equation using the quadratic formula:

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

First, let's compute the discriminant:

[Delta b^2 - 4ac (-23)^2 - 4(3)(-110) 529 1320 1849]

Next, we will solve for x:

[x frac{23 pm sqrt{1849}}{6} frac{23 pm 43}{6}]

Thus, the solutions are:

[x frac{23 43}{6} frac{66}{6} 11] [x frac{23 - 43}{6} frac{-20}{6} -frac{10}{3}]

Solution by Factoring

Another method to solve the equation (3x^2 - 23x - 110 0) is by factoring.

We can rewrite the equation by recognizing that 3x^2 - 23x - 110 can be factored into the form:

[3x^2 - 33x 1 - 110 0]

Grouping terms:

[(3x^2 - 33x) (1 - 110) 0]

Factoring out the common terms:

[3x(x-11) 10(x-11) 0]

Combining terms:

[(3x 10)(x - 11) 0]

Setting each factor equal to zero gives us the solutions:

[(3x 10) 0 Rightarrow x -frac{10}{3}] [(x - 11) 0 Rightarrow x 11]

Solution by Completing the Square

The general form for completing the square is:

[ax^2 bx c 0]

Starting with the equation (3x^2 - 23x - 110 0), we divide by the coefficient of (x^2), which is 3:

[x^2 - frac{23}{3}x - frac{110}{3} 0]

We then move the constant term to the right side:

[x^2 - frac{23}{3}x frac{110}{3}]

To complete the square, add and subtract (left(frac{23}{6}right)^2": [(x - frac{23}{6})^2 - frac{529}{36} - frac{110}{3} 0]

Simplify the right side:

[(x - frac{23}{6})^2 frac{1799}{36}]

Solving for x:

[x - frac{23}{6} pm frac{43}{6}]

Thus, the solutions are:

[x frac{23 43}{6} 11] [x frac{23 - 43}{6} -frac{10}{3}]

Conclusion

By using different methods—quadratic formula, factoring, and completing the square—it is apparent that the solutions to the equation (3x^2 - 23x - 110 0) are (x 11) and (x -frac{10}{3}). Each method provides a unique insight and reinforces the principles of algebra and equation solving.

Keywords: quadratic equation, solving quadratic equations, methods to solve quadratic equations