Problem 68 - Elementary Algebra

Question

For the following problems, solve the equations, if possible. $$ a^{2}+2 a-3=0 $$

Step-by-Step Solution

Answer

Answer: The roots of the quadratic equation are a = 1 and a = -3.

Step 1: Identify the coefficients

In the given quadratic equation, we can identify the coefficients as follows: A quadratic equation has the form $$ax^2 + bx + c = 0$$ Comparing this with our given equation $$a^2 + 2a - 3 = 0$$, we get: a=1, b=2, and c=-3.

Step 2: Calculate the discriminant

The discriminant D of a quadratic equation with coefficients a, b, and c is given by the formula: $$D = b^2 - 4ac$$ Now, plug in the values of a, b, and c from our equation: $$D = (2)^2 - 4(1)(-3) = 4 + 12 = 16$$

Step 3: Use the quadratic formula to find the roots

The quadratic formula for solving the roots of a quadratic equation with coefficients a, b, and c is given by: $$x = \frac{-b \pm \sqrt{D}}{2a}$$ Plug in the values of a, b, and D from our previous steps: $$a = \frac{-2 \pm \sqrt{16}}{2(1)}$$

Step 4: Simplify and find the two roots

Now, we can simplify our expression to find the two roots: $$a_1 = \frac{-2 + \sqrt{16}}{2} = \frac{-2 + 4}{2} = \frac{2}{2} = 1$$ $$a_2 = \frac{-2 - \sqrt{16}}{2} = \frac{-2 - 4}{2} = \frac{-6}{2} = -3$$ So, the two roots of the given quadratic equation are a = 1 and a = -3.

Key Concepts

Discriminant

The discriminant is a crucial component of quadratic equations. It helps us determine the nature of the roots without having to find them directly. For a quadratic equation given by the standard form \[ \ ax^2 + bx + c = 0 \] The discriminant is represented as: \[ \ D = b^2 - 4ac \] Here is what the discriminant tells us:

• If \(D > 0\), the equation has two distinct real roots.
• If \(D = 0\), the equation has exactly one real root, meaning the roots are equal or repeated.
• If \(D < 0\), the equation has no real roots, but two complex roots.

By using the discriminant, we can avoid unnecessary calculations if we only need to know the root nature. In our example, the discriminant was calculated as 16. Since 16 is greater than zero, it indicates there are two real and distinct roots for the equation \( a^2 + 2a - 3 = 0 \). This insight is vital before proceeding with solving the equation.

Quadratic Formula

The quadratic formula provides a straightforward method to calculate the roots of any quadratic equation, as long as the coefficients are known. The formula is given by: \[ \ x = \frac{-b \pm \sqrt{D}}{2a} \] Where \(D\) is the discriminant calculated as \(b^2 - 4ac\). This formula not only simplifies the solving process but ensures that all possible roots are found. Here’s how it works:

• Calculate the discriminant \(D\) using known values of \(a\), \(b\), and \(c\).
• Substitute \(b\), \(D\), and \(a\) into the quadratic formula.
• Solve for \(x\) using addition \((+)\) and subtraction \((-\)) with the square root term. This gives you the two potential roots.

In the step-by-step solution, the quadratic formula was used after calculating the discriminant. With \(D = 16\), the roots \(a\) yielded were 1 and -3. Thus, the formula's power is seen in its ability to disclose both possible solutions for \(a\).

Roots of Quadratic Equation

The roots of a quadratic equation are the values of the variable that satisfy the equation, meaning they make the equation true (equal to zero). When solving, you must first express the equation in the standard quadratic form: \[ \ ax^2 + bx + c = 0 \] To find these roots:

• Evaluate the discriminant to establish the number and type of roots.
• Use the quadratic formula to find the precise root values.

In the example equation \(a^2 + 2a - 3 = 0\), the roots were found to be 1 and -3. These roots represent the points where the parabola described by the quadratic equation intersects the x-axis in a standard graph. Finding these intersections is at the heart of quadratic problem-solving, as it provides the solution sets in real-world applications, such as calculating trajectories or optimizing areas. Understanding roots elucidates the behavior of quadratic functions and their graphical representations.