Problem 59 - Intermediate Algebra

Question

For each quadratic equation, first use the discriminant to determine whether the equation has two nonreal complex solutions, one real solution with a multiplicity of two, or two real solutions. Then solve the equation. $$ 3 x^{2}+4 x=2 $$

Step-by-Step Solution

Answer

The equation has two real solutions: \( x = \frac{-2 + \sqrt{10}}{3} \) and \( x = \frac{-2 - \sqrt{10}}{3} \).

Step 1: Rewrite the Equation in Standard Form

Start by rewriting the equation in its standard quadratic form, \( ax^2 + bx + c = 0 \). For our equation, subtract 2 from both sides to obtain \( 3x^2 + 4x - 2 = 0 \). Here, \( a = 3 \), \( b = 4 \), and \( c = -2 \).

Step 2: Calculate the Discriminant

The discriminant, \( \Delta \), for a quadratic equation \( ax^2 + bx + c = 0 \) is given by \( \Delta = b^2 - 4ac \). Substitute the values from the equation: \( \Delta = 4^2 - 4 \cdot 3 \cdot (-2) = 16 + 24 = 40 \).

Step 3: Determine the Nature of the Roots

Interpret the discriminant: \( \Delta = 40 \) is positive, indicating that the equation has two distinct real solutions.

Step 4: Solve the Quadratic Equation Using the Quadratic Formula

Use the quadratic formula: \[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \]. Substitute \( a = 3 \), \( b = 4 \), and \( \Delta = 40 \):\[ x = \frac{-4 \pm \sqrt{40}}{6} \].Simplify \( \sqrt{40} \) to \( 2\sqrt{10} \), so:\[ x = \frac{-4 \pm 2\sqrt{10}}{6} \].Reduce this expression to:\[ x = \frac{-2 \pm \sqrt{10}}{3} \].

Key Concepts

Discriminant

In quadratic equations, the discriminant is a key element that helps determine the type of solutions we can expect. It's a component of the quadratic equation's standard form: \( ax^2 + bx + c = 0 \). The discriminant \( \Delta \) is calculated using the formula:

• \( \Delta = b^2 - 4ac \)

By calculating this value, you can understand the nature of the solutions without even solving the equation. Here's how it works:

• If \( \Delta > 0 \), there are two distinct real solutions.
• If \( \Delta = 0 \), there is exactly one real solution with a multiplicity of two. This means the parabola just touches the x-axis at one point.
• If \( \Delta < 0 \), the solutions are two non-real complex numbers.

Understanding the discriminant is crucial, as it guides us on whether to proceed with solving for real or complex solutions.

Real Solutions

Real solutions in quadratic equations result from cases where the discriminant is zero or positive. As discussed, the discriminant \( \Delta \) helps in predicting whether these solutions will exist or not.

• When \( \Delta > 0 \), it indicates two distinct and different real solutions. This happens because the quadratic touches the x-axis at two different points.
• When \( \Delta = 0 \), it suggests there is one real solution. This means the graph of the quadratic equation just skims the x-axis, creating what's known as a repeated or "double root."

With real solutions, the values calculated will be numbers that can be plotted on a real number line, giving clear points of intersection on the x-axis if graphed. Therefore, when determining the method to solve a quadratic equation, knowing the type of real solutions helps decide if further factorizations or specific formulas are needed.

Quadratic Formula

Solving quadratic equations can often be straightforward with the quadratic formula. The formula provides a way to find the roots of any quadratic equation \( ax^2 + bx + c = 0 \) and is written as:

• \( x = \frac{-b \pm \sqrt{\Delta}}{2a} \)

where \( \Delta \) is the discriminant. The quadratic formula works effectively for all types of solutions—real or complex.

• For two distinct real solutions, both \( x \) values, derived using the "\( \pm \)" part of the formula, are actual real numbers.
• For the single real solution when \( \Delta = 0 \), the formula simplifies as both \( x \) values calculate to the same number, confirming only one point of contact with the x-axis.
• For non-real complex solutions, when \( \Delta < 0 \), the formula neatly incorporates imaginary numbers through the square root of a negative determinant, leading to a complex solution set involving part real and part imaginary numbers.

Thus, the quadratic formula is a robust and reliable method that handles all scenarios presented by quadratic equations.