Solving the Quadratic Equation 2x^2 - 3x - 20 0 with the Quadratic Formula

When faced with the quadratic equation 2x^2 - 3x - 20 0, you can solve it effectively using the quadratic formula. The quadratic formula is a powerful tool for solving quadratic equations of the form ax^2 bx c 0. The formula is given by:

The Quadratic Formula

The general formula is:

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

For the given quadratic equation 2x^2 - 3x - 20 0, the coefficients are a 2, b -3, and c -20.

Step-by-Step Solution

First, identify the coefficients: a 2, b -3, and c -20.

Calculate the discriminant (b^2 - 4ac):

(-3)^2 - 4(2)(-20) 9 160 169

The discriminant is 169, which is a positive number, indicating that there are two real and distinct solutions.

Apply the quadratic formula:

Solve for x_1:

x_1 frac{-(-3) sqrt{169}}{2(2)} frac{3 13}{4} frac{16}{4} 4

Solve for x_2:

x_2 frac{-(-3) - sqrt{169}}{2(2)} frac{3 - 13}{4} frac{-10}{4} -frac{5}{2}

Thus, the solutions to the equation 2x^2 - 3x - 20 0 are:

x_1 4 x_2 -frac{5}{2}

Alternative Factoring Method

For further clarification, let's also solve the same equation using factoring. This method may not always be straightforward, but it can be useful for simpler equations.

Factoring the Equation

Let's consider:

Start by factoring the equation:

2x^2 - 3x - 20 2x^2 - 8x 5x - 20

Next, factor by grouping:

2x^2 - 8x 5x - 20 2x(x - 4) 5(x - 4)

Factor out the common term:

(2x 5)(x - 4)

Set each factor equal to zero and solve for x:

2x 5 0 ? x -frac{5}{2} x - 4 0 ? x 4

Conclusion

The solutions to the quadratic equation 2x^2 - 3x - 20 0 are x 4 and x -frac{5}{2}. You can verify these solutions by substituting them back into the original equation.