Problem 54 - Algebra 1

Question

solve the quadratic equation. $$x^{2}-4 x-15=0$$

Step-by-Step Solution

Answer

The solutions to the equation \(x^{2}-4x-15=0\) are \(x = 2 + \sqrt{19}\) and \(x = 2 - \sqrt{19}\)

Step 1: Identify the coefficients a, b, and c

From the quadratic equation \(x^{2}-4x-15=0\), the coefficients \(a\), \(b\), and \(c\) are as follows: \(a = 1\), \(b = -4\), and \(c = -15\).

Step 2: Substitute the coefficients into the quadratic formula

The quadratic formula is \(\frac{-b ± \sqrt{b^{2}-4ac}}{2a}\). Substituting the values of \(a\), \(b\), and \(c\) we get: \(\frac{4 ± \sqrt{(-4)^{2}-4(1)(-15)}}{2(1)}\)

Step 3: Simplify the equation

Simplify the equation to find the roots of the quadratic: \(\frac{4 ± \sqrt{16+60}}{2}\), which simplifies to \(\frac{4 ± \sqrt{76}}{2}\) or \(\frac{4 ± 2\sqrt{19}}{2}\)

Step 4: Solve the equation

Finally, solving the equation gives us the roots: \(x = \frac{4 + 2\sqrt{19}}{2} = 2 + \sqrt{19}\) and \(x = \frac{4 - 2\sqrt{19}}{2} = 2 - \sqrt{19}\)

Key Concepts

Quadratic Formula

The quadratic formula is a handy tool in mathematics for finding the roots of any quadratic equation. Quadratic equations are polynomial equations of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. Here, \( a \) must be non-zero, meaning the equation indeed contains an \( x^2 \) term. The quadratic formula is written as:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This formula provides solutions for the variable \( x \) by substituting the values of \( a \), \( b \), and \( c \). The term under the square root, \( b^2 - 4ac \), is known as the discriminant. It is crucial because it determines the nature and the number of the roots of the equation. If you ever need to solve a quadratic equation, remember this powerful formula as it guarantees finding the roots as long as you know your coefficients.

Roots of a Quadratic Equation

Understanding the roots of a quadratic equation is essential as they represent the values of \( x \) that satisfy the equation. The roots can be real or complex numbers, and they serve as the points where the quadratic graph intersects the x-axis.
The number and nature of the roots are primarily determined by the discriminant \( b^2 - 4ac \):

• If the discriminant is positive, there are two distinct real roots.
• If the discriminant is zero, there is exactly one real root, also known as a repeated or double root.
• If the discriminant is negative, the roots are complex and occur as a conjugate pair.

In the context of our example, the discriminant is positive \( b^2 - 4 \cdot 1 \cdot (-15) = 76 \), indicating two distinct real roots. This tells us that the parabolic graph of \( x^2 - 4x - 15 \) crosses the x-axis at two points.

Solving Quadratic Equations

Solving quadratic equations involves finding their roots, which are the values that make the equation equal zero. There are multiple methods to solve quadratic equations, but using the quadratic formula is one of the most straightforward techniques, especially when factoring seems less feasible or when the equation involves large numbers.
Here's a streamlined approach to solving using the quadratic formula:

• Identify the coefficients \( a \), \( b \), and \( c \) from the equation, ensuring that it is in standard form.
• Substitute these coefficients into the quadratic formula.
• Simplify the solution step-by-step, carefully following mathematical rules to handle arithmetic operations and square roots correctly.
• Arrive at the solution, presenting it either as exact roots or as decimal approximations, depending on the problem's requirements.

Using our example, we substituted \( a = 1 \), \( b = -4 \), and \( c = -15 \) into the formula and simplified the results to find the roots: \( x = 2 + \sqrt{19} \) and \( x = 2 - \sqrt{19} \). These solutions confirm that solving a quadratic equation is both systematic and efficient with the use of the quadratic formula.