Problem 31 - Algebra 1: Concepts and Skills

Question

Determine whether the equation has two solutions, one solution, or no real solution. (Lesson 9.7) $$ 2 x^{2}+4 x-2=0 $$

Step-by-Step Solution

Answer

The equation \(2x^{2}+4x-2=0\) has two real solutions.

Step 1: Identify a, b and c

In the given quadratic equation \(2x^{2}+4x-2=0\), the coefficients are as follows: \(a=2\), \(b=4\), and \(c=-2\).

Step 2: Calculate the discriminant

The discriminant is calculated using the formula \(b^{2}-4ac\). By substituting the values of \(a\), \(b\), and \(c\), the discriminant is found to be \(4^{2} - 4*2*(-2) =16 +16 = 32\).

Step 3: Determine the number of solutions

Since the discriminant is positive (32>0), the given quadratic equation has two real solutions.

Key Concepts

Discriminant

In the world of quadratic equations, the discriminant is a very useful tool to determine the nature of solutions that the equation might have. It comes from the quadratic formula, which is used to find the roots of a quadratic equation. The formula for the discriminant, denoted as \( \Delta \), is \( b^2 - 4ac \).
Here:

• \( a \) is the coefficient of \( x^2 \)
• \( b \) is the coefficient of \( x \)
• \( c \) is the constant term

By calculating the discriminant, you can predict how many real solutions the quadratic equation has without actually solving the equation completely. If you find a positive discriminant, it means you'll have two distinct real solutions. A discriminant of zero indicates exactly one real solution, as the roots coincide. If it's negative, the equation has no real solutions but two complex ones.
So, remember, just a quick calculation of \( b^2 - 4ac \) can carry a lot of information about the nature of the roots of a quadratic equation.

Real Solutions

Real solutions of a quadratic equation are the x-values where the graph of the equation crosses the x-axis. For quadratic equations in the form \( ax^2 + bx + c = 0 \), the solutions can be identified using the discriminant.
Here's a simple guide:

• If the discriminant \( b^2 - 4ac \) is greater than zero, the equation has two distinct real solutions. This happens because the parabola intersects the x-axis at two points.
• When the discriminant is exactly zero, there is exactly one real solution. This is due to the parabola just touching the x-axis at its vertex, known as a repeated real root.
• Finally, if the discriminant is less than zero, the equation lacks real solutions because the parabola does not cross the x-axis at all, resulting in complex roots instead.

Understanding the nature of real solutions through the discriminant helps simplify analyzing quadratic equations. You can envision how the parabola behaves with respect to the x-axis by simply calculating this critical value.

Quadratic Formula

The Quadratic formula is a powerful tool for solving quadratic equations, which are polynomials of the second degree. These equations take the form \( ax^2 + bx + c = 0 \), and to find the values of \( x \) that satisfy this equation, you can use the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula provides a straightforward way to calculate the roots of a quadratic equation, no matter how complex it might seem at first glance. Let's break down how it works:

• The \(b^2 - 4ac\) part, known as the discriminant, indicates the nature and number of the roots as discussed earlier.
• The \( \pm \) sign signifies that there are generally two solutions: one that adds the square root of the discriminant and one that subtracts it, reflecting the potential for two roots.
• The denominator \( 2a \) ensures that the variable \( x \) is correctly isolated on one side of the equation.

Using the quadratic formula ensures you can solve any quadratic equation by simply plugging in the values of \( a \), \( b \), and \( c \), deriving precise solutions accurately every time. It's essential to become familiar with this formula, as it is a fundamental aspect of algebra.