Problem 25 - Algebra 1: Concepts and Skills

Question

Determine whether the equation has two solutions, one solution, or no real solution. \(2 x^{2}+3 x-2=0\)

Step-by-Step Solution

Answer

The quadratic equation has two real solutions.

Step 1: Identify a, b, and c from the equation

The quadratic equation is in the form \(ax^2 + bx + c = 0\). Here \(a = 2\), \(b = 3\), and \(c = -2\) from the given equation \(2x^2 + 3x - 2 = 0\).

Step 2: Compute the Discriminant

The discriminant is given by \(D = b^2 - 4ac\). Substituting the values of \(a\), \(b\), and \(c\) we have \(D = (3)^2 - 4 * 2 * (-2) = 9 + 16 = 25\)

Step 3: Determine the number of real solutions

Since the discriminant \(D = 25\) is greater than 0, it means the given quadratic equation \(2x^2 + 3x - 2 = 0\) has two real solutions.

Key Concepts

Understanding the Discriminant

The discriminant is a crucial component in the study of quadratic equations. It helps us understand the nature of the roots, or solutions, of a quadratic equation. A quadratic equation typically looks like this: \[ ax^2 + bx + c = 0 \] where \(a\), \(b\), and \(c\) are constants. To find the discriminant \(D\), you use the formula:\[ D = b^2 - 4ac \]

• If the discriminant is greater than zero: The equation has two distinct real solutions.
• If the discriminant equals zero: The quadratic equation has exactly one real solution, as the roots are coincident, or the same.
• If the discriminant is less than zero: The equation has no real solutions, implying that the roots are complex numbers.

By calculating the discriminant, we are simply examining the part of the quadratic formula that determines how the values under the square root sign behave. Knowing the value of \(D\) gives immediate information without having to solve the equation fully.

Number of Solutions Determined by the Discriminant

Using the discriminant, we can easily determine the number of solutions a quadratic equation will have. As a restatement, consider the quadratic formula derived from the general equation \(ax^2 + bx + c = 0\): \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]The term under the square root, \(b^2 - 4ac\), is the discriminant.

• When \(D > 0\): Since the square root of a positive number is real and yields two solutions due to the \(\pm\) sign, there are two solutions.


• When \(D = 0\): The square root of zero is zero, yielding a single solution, \(x = -b/2a\). This is when the vertex of the parabola touches the x-axis.


• When \(D < 0\): A negative discriminant under a square root results in an imaginary number, indicating no real solutions exist for the equation.

Thus, examining the discriminant is a quick way to gauge how many real roots the quadratic equation will have without fully solving it.

What Real Solutions Mean in Context

"Real solutions" refer to the values of \(x\) that satisfy the quadratic equation and are real numbers. When dealing with real solutions, we mean that the solution points lie on the real number line. In practical terms:

• Two real solutions (\(D > 0\)): The graph of the equation, a parabola, intersects the x-axis at two distinct points. Each intersection is a solution or root of the equation.


• One real solution (\(D = 0\)): The parabola just touches the x-axis at exactly one point—this is called a double root because it's both a maximum or a minimum point.


• No real solutions (\(D < 0\)): The parabola does not intersect the x-axis at all, indicating that no actual, recommendable number satisfies the equation in terms of real solutions.

Understanding when and why solutions are considered real is vital when assessing graphs and predictions based on quadratic functions. Real solutions can often mean feasible outcomes in practical applications such as physics, engineering, or economics.