Problem 16 - Algebra 2 and Trigonometry

Question

In \(15-23 :\) a. Find the value of the discriminant and determine if the roots of the quadratic equation are \((1)\) rational and unequal, \((2)\) rational and equal, \((3)\) irrational and unequal, or \((4)\) not real numbers. b. Use any method to find the real roots of the equation if they exist. $$ 2 x^{2}+7 x=0 $$

Step-by-Step Solution

Answer

The roots are real and unequal: \(x = 0\) and \(x = -\frac{7}{2}\).

Step 1: Identify the Quadratic Equation Components

For the quadratic equation in the form \(ax^2 + bx + c = 0\), identify the values of \(a\), \(b\), and \(c\). From \(2x^2 + 7x = 0\), we have \(a = 2\), \(b = 7\), and \(c = 0\).

Step 2: Calculate the Discriminant

The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is given by \(D = b^2 - 4ac\). Plugging in the identified values, we have \(D = 7^2 - 4 \cdot 2 \cdot 0 = 49 - 0 = 49\).

Step 3: Determine the Nature of the Roots

Based on the value of the discriminant: - If \(D > 0\), roots are real and unequal.- If \(D = 0\), roots are real and equal.- If \(D < 0\), roots are not real.Here, \(D = 49 > 0\), so the roots are real and unequal.

Step 4: Solve for Real Roots

Since real roots exist (from Step 3, \(D > 0\)), you can solve for \(x\) using factorization or the quadratic formula.The equation \(2x^2 + 7x = 0\) can be factored as \(x(2x + 7) = 0\).Setting each factor equal to zero gives: \[ x = 0 \] \[ 2x + 7 = 0 \Rightarrow 2x = -7 \Rightarrow x = -\frac{7}{2} \]

Step 5: Confirm the Solution

Verify the results by plugging \(x = 0\) and \(x = -\frac{7}{2}\) back into the original equation:- For \(x = 0\): \(2(0)^2 + 7(0) = 0\), which is true.- For \(x = -\frac{7}{2}\): Substitute into \(2x^2 + 7x = 0\), confirm it holds true as well.Both values satisfy the equation.

Key Concepts

The Discriminant

When talking about quadratic equations, the discriminant plays a key role. The discriminant is derived from the quadratic formula and tells us about the types of solutions we can expect. For a quadratic equation in the form \( ax^2 + bx + c = 0 \), the discriminant \( D \) is calculated using the formula \( D = b^2 - 4ac \).
It's important to remember that the discriminant does not give you the solution directly, but it indicates what type of roots the equation will have. Let's break it down further with different scenarios:

• If \( D > 0 \): There are two real and unequal roots.
• If \( D = 0 \): There is one real root, effectively a double root (they are equal).
• If \( D < 0 \): There are no real roots; the solutions are complex numbers.

By calculating the discriminant first, you’ll know exactly what kind of solutions to expect before you even start solving the equation. In our example, \( D = 49 \), which tells us the roots are real and unequal, setting the stage for factorizations.

Finding Real Roots

Once we know the discriminant is positive, indicating real solutions, the next step is finding the actual values of these roots. Two common methods are the quadratic formula and factorization, but since the problem allows any method when \( D > 0 \), we'll focus on factorization here.
The original equation is \( 2x^2 + 7x = 0 \). Notice this can be factored by taking out a common factor, \( x \), resulting in \( x(2x + 7) = 0 \). This gives us two simple equations to solve:

• \( x = 0 \)
• \( 2x + 7 = 0 \)

The first equation is straightforward, leading straight to a root of \( x = 0 \). Solving the second equation for \( x \) gives:- \( 2x = -7 \)- \( x = -\frac{7}{2} \)
These are the real roots of the quadratic equation. They correspond directly to the factorization used, confirming their validity.

Factorization in Solving Quadratics

Factorization is one of the simplest methods to solve quadratic equations when the equation is factorable. This process allows us to express the quadratic in a product of simpler polynomials, which can be solved individually. Here's why factorization works so well:
Starting from the equation \( ax^2 + bx + c = 0 \), if it can be expressed as \( (px + q)(rx + s) = 0 \), we can determine the roots easily. The idea is based on the zero-product property, which says if a product is zero, at least one of the factors must be zero.
For individuals who prefer straightforward calculations or when the coefficients lend themselves well to division, factorization can be especially useful. In our case, recognizing \( 2x \) and \( 7 \) as terms that can factor out led to simple solutions. In factored form \( x(2x + 7) = 0 \), solving for \( x \) requires almost no added steps once factorization is complete, making it efficient for finding roots. Plus, verifying is as easy as substituting the roots back into the original equation.

Understanding the Nature of Roots

The nature of the roots is the term used to describe what kind of solutions you have after solving a quadratic equation. Much of this nature is told by the discriminant, but seeing the final form of the roots connects it all together.
First, consider the three possibilities earlier stated from the discriminant:

• Two real and distinct roots when \( D > 0 \)
• One real, double root when \( D = 0 \)
• Two complex roots when \( D < 0 \)

For \( D > 0 \), the roots are not just real, they're distinct, meaning you won't find a repeat value. This uniqueness is confirmed through factorization or any other solving method.
In our original case, identifying that \( x = 0 \) and \( x = -\frac{7}{2} \) are the roots makes clear the distinctness: each is a unique solution to the equation \( 2x^2 + 7x = 0 \). The understanding becomes that the nature tells you not only the "realness" of the roots but also their characteristics—whether they repeat or are part of complex pairs.