Problem 57 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ 2 x^{2}-3 x=20 $$

Step-by-Step Solution

Answer

The solutions are \( x = 4 \) and \( x = -2.5 \).

Step 1: Write the equation in standard form

The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \). Start by moving all terms to one side of the equation so that it equals zero: \( 2x^2 - 3x - 20 = 0 \).

Step 2: Identify coefficients

Determine the values of \( a \), \( b \), and \( c \) from the equation \( 2x^2 - 3x - 20 = 0 \). Here, \( a = 2 \), \( b = -3 \), and \( c = -20 \).

Step 3: Use the quadratic formula

The quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), allows us to find the roots of a quadratic equation. Substitute the coefficients: \( x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 2 \cdot (-20)}}{2 \cdot 2} \).

Step 4: Simplify under the square root

Calculate the discriminant \( b^2 - 4ac \): \( (-3)^2 - 4 \cdot 2 \cdot (-20) = 9 + 160 = 169 \).

Step 5: Solve for x

Substitute the discriminant back into the quadratic formula: \( x = \frac{3 \pm \sqrt{169}}{4} \), which simplifies to \( x = \frac{3 \pm 13}{4} \).

Step 6: Find the individual solutions

Compute \( x = \frac{3 + 13}{4} = 4 \) and \( x = \frac{3 - 13}{4} = -2.5 \).

Step 7: Verify the solutions

Substitute \( x = 4 \) and \( x = -2.5 \) back into the original equation to ensure they satisfy \( 2x^2 - 3x = 20 \). Both solutions satisfy the equation.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). This formula helps you find the values of \( x \) that make the equation true. It is especially useful when the quadratic does not factor easily or when you're looking for an algebraic solution quickly. The formula is given by:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This might seem complex at first, but it's essentially just a recipe where you plug in the values of \( a \), \( b \), and \( c \) from your equation. For example, in the equation \( 2x^2 - 3x - 20 = 0 \), \( a = 2 \), \( b = -3 \), and \( c = -20 \). Substitute these into the quadratic formula and simplify.

Remember, the "\( \pm \)" symbol means you will get two solutions: one by adding and one by subtracting.

Discriminant

The discriminant is a key part of the quadratic formula, found under the square root sign: \( b^2 - 4ac \). Knowing its value can tell us a lot about the solutions of the quadratic equation without even needing to calculate them exactly.

• If the discriminant is positive, \( b^2 - 4ac > 0 \), the quadratic equation has two distinct real roots.
• If it's zero, \( b^2 - 4ac = 0 \), the equation has exactly one real root (or a repeated root).
• If the discriminant is negative, \( b^2 - 4ac < 0 \), the roots are complex, meaning they aren't real numbers.

In our example, the discriminant \((9 + 160 = 169)\) is positive, indicating that there are two distinct real solutions. Calculating the discriminant first can give you insight into the nature of solutions before you've completed all the math.

Standard Form of Quadratic Equation

The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \). Understanding this format helps you easily identify the coefficients \( a \), \( b \), and \( c \), which are essential for using the quadratic formula or other methods of solving quadratic equations.

Start by ensuring your quadratic has all terms on one side of the equation, equating to zero. This may involve rearranging or simplifying your original expression. In our case, we start with \( 2x^2 -3x = 20 \) and adjust it to \( 2x^2 - 3x - 20 = 0 \).

Now, identify each coefficient easily:

• \( a = 2 \), the coefficient of \( x^2 \)
• \( b = -3 \), the coefficient of \( x \)
• \( c = -20 \), the constant term

Identifying these values correctly is crucial. It allows you to accurately apply the quadratic formula and find the solutions to the equation.