Solving the Quadratic Equation 4x2 19x - 5 0 with the Quadratic Formula

The quadratic equation is a fundamental concept in algebra, used to find the solutions of equations of the form ax^2 bx c 0. The quadratic formula is a powerful tool for solving such equations. Letrsquo;s explore how to solve the equation 4x^2 19x - 5 0 using the quadratic formula.

Given the quadratic equation:

4x^2   19x - 5  0

The quadratic formula is given by:

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

where a 4, b 19, and c -5.

Step-by-Step Solution

Identify Coefficients: First, identify the coefficients a, b, and c in the quadratic equation 4x^2 19x - 5 0. Substitute into the Quadratic Formula: Substitute a 4, b 19, and c -5 into the quadratic formula. Calculate the Discriminant: Calculate the discriminant Δ b^2 - 4ac. Find the Roots: Use the quadratic formula to find the roots of the equation.

Calculation Details

Identify Coefficients: a 4, b 19, c -5 Substitute into the Quadratic Formula: Calculate the Discriminant:

Calculation Process

Discriminant:

Δ 19^2 - 4(4)(-5) 361 80 441

Apply the Quadratic Formula:

x frac{-19 pm sqrt{441}}{2(4)} frac{-19 pm 21}{8}

Find the Roots:

x_1 frac{-19 21}{8} frac{2}{8} frac{1}{4}

x_2 frac{-19 - 21}{8} frac{-40}{8} -5

Final Answer

The solutions to the quadratic equation 4x^2 19x - 5 0 are:

x 1/4 x -5

Additional Notes

Note that this quadratic equation can also be factored directly. The factors are:

(4x - 1)(x   5)  0

Setting each factor to zero gives:

4x - 1  0 quad text{or} quad x   5  0

Which simplifies to:

x  frac{1}{4} quad text{or} quad x  -5

Thus, both methods yield the same solutions: x 1/4 and x -5.

Conclusion

The quadratic formula provides a systematic approach to solving quadratic equations of the form ax^2 bx c 0. While the direct factoring method is quicker for simple cases, the quadratic formula is a reliable method that works for all quadratic equations.