Problem 49 - Algebra A Combined Function

Question

Use the discriminant to determine the number and types of solutions of each equation. $$ 9 x-2 x^{2}+5=0 $$

Step-by-Step Solution

Answer

The equation has two distinct real solutions.

Step 1: Identify the coefficients

We need to rewrite the equation in standard quadratic form, which is \(ax^2 + bx + c = 0\). For the equation \(9x - 2x^2 + 5 = 0\), rearrange to get \(-2x^2 + 9x + 5 = 0\). Here, the coefficients are \(a = -2\), \(b = 9\), and \(c = 5\).

Step 2: Write the formula for the discriminant

The discriminant \(D\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula \(D = b^2 - 4ac\).

Step 3: Substitute the coefficients into the discriminant formula

Using the formula \(D = b^2 - 4ac\), substitute \(a = -2\), \(b = 9\), and \(c = 5\). This gives us: \(D = 9^2 - 4(-2)(5)\).

Step 4: Calculate the discriminant

Perform the calculations: \(9^2 = 81\) and \(-4(-2)(5) = 40\). Therefore, \(D = 81 + 40 = 121\).

Step 5: Determine the number and type of solutions

Since the discriminant \(D = 121\) is greater than zero, the quadratic equation has two distinct real solutions.

Key Concepts

Solution of Quadratic Equations

To solve a quadratic equation, we first need to identify its standard form, which is expressed as \(ax^2 + bx + c = 0\). This structure is key because it allows us to use various techniques to find the values of \(x\), known as the solutions or roots. These solutions are the points where the graph of the equation crosses the x-axis. When solving quadratics, you'll typically encounter different methods:

• Factoring, which requires the quadratic to be easily decomposable into simple binomials.
• Completing the square, a method to rewrite the equation in a perfect square form.
• The quadratic formula, a universal technique that works for any quadratic equation, which we'll discuss next.

Remember, correctly identifying the coefficients \(a\), \(b\), and \(c\) from the equation is the first step. This is crucial because these values will guide the entire solving process.

Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions of any quadratic equation. It is derived from the standard form \(ax^2 + bx + c = 0\) and is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula directly incorporates the discriminant, \(b^2 - 4ac\), under the square root. Here's how to apply the quadratic formula:

• First, accurately identify the coefficients \(a\), \(b\), and \(c\) from the standard form of the equation.
• Calculate the discriminant, \(b^2 - 4ac\), since it determines the nature of the roots.
• Substitute these values into the formula to find \(x\).

Remember, "\(\pm\)" signifies that there are two possible solutions: one using the plus sign and the other using the minus sign. This is why a quadratic equation can have up to two roots.

Real and Distinct Solutions

The discriminant, \(D = b^2 - 4ac\), is a crucial determinant in identifying the nature of the solutions of a quadratic equation. Here's why it's important:A positive discriminant (\(D > 0\)) indicates that the quadratic equation has two real and distinct solutions. This means the graph of the quadratic equation intersects the x-axis at two unique points. In our example, the discriminant was calculated to be 121, which is positive and thus confirms two distinct real roots.This contrasts with:

• A zero discriminant (\(D = 0\)), which results in exactly one real solution or a double root. Here, the graph just touches the x-axis.
• A negative discriminant (\(D < 0\)), which signifies no real solutions, meaning the graph does not cross the x-axis.

Understanding the discriminant helps predict how many x-intercepts there will be, thus providing insight into the solution behavior of quadratic equations.