Problem 51 - Elementary Algebra

Question

Solve using any method. $$ x 2+4 x-60=0 $$

Step-by-Step Solution

Answer

The solutions are \( x = 6 \) and \( x = -10 \).

Step 1: Write the Equation in Standard Form

The given equation is already in standard quadratic form, which is \( x^2 + 4x - 60 = 0 \). In this form, we can identify the coefficients as \( a = 1 \), \( b = 4 \), and \( c = -60 \).

Step 2: Calculate the Discriminant

The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is given by the formula \( \Delta = b^2 - 4ac \). Substitute the values of \( a \), \( b \), and \( c \) into the formula:\[\Delta = 4^2 - 4 \times 1 \times (-60) = 16 + 240 = 256.\]Since \( \Delta = 256 > 0 \), the equation has two distinct real roots.

Step 3: Apply the Quadratic Formula

The quadratic formula \( x = \frac{{-b \pm \sqrt{\Delta}}}{{2a}} \) can now be used to find the roots of the equation.Substitute \( a = 1 \), \( b = 4 \), and \( \Delta = 256 \) into the formula:\[x = \frac{{-4 \pm \sqrt{256}}}{{2 \times 1}} = \frac{{-4 \pm 16}}{{2}}.\]Simplify to find the roots.

Step 4: Solve for Each Root

Calculate each root separately:1. \( x_1 = \frac{{-4 + 16}}{2} = \frac{12}{2} = 6. \)2. \( x_2 = \frac{{-4 - 16}}{2} = \frac{-20}{2} = -10. \)Thus, the solutions to the equation are \( x = 6 \) and \( x = -10 \).

Key Concepts

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. A quadratic equation is typically in the form:

• \( ax^2 + bx + c = 0 \)

where \( a \), \( b \), and \( c \) are constants with \( a eq 0 \). The quadratic formula itself is written as:

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

This formula allows you to find the values of \( x \) that satisfy the quadratic equation. The main advantage is that it can be used for any quadratic equation, no matter how simple or complex. By plugging in the values of \( a \), \( b \), and \( c \) from your equation, you can quickly solve for \( x \). In our case, since \( a = 1 \), \( b = 4 \), and \( c = -60 \), we substitute these into the formula to find the roots of the given quadratic equation.

Understanding the Discriminant

The discriminant is a crucial part of the quadratic formula, found within the square root:

• \( \Delta = b^2 - 4ac \)

It helps in determining the nature of the roots of the quadratic equation:

• If \( \Delta > 0 \), there are two distinct real roots.
• If \( \Delta = 0 \), there is exactly one real root (also called a repeated or double root).
• If \( \Delta < 0 \), there are no real roots, but two complex roots.

In our original problem, we calculated \( \Delta \) and found it to be 256. Since 256 is positive, we concluded that our quadratic equation has two distinct real roots. This tells us a lot about the behavior of the equation's graph, which crosses the x-axis at two points.

What Are Real Roots?

Real roots are the solutions to the quadratic equation that you can plot on a number line. When we talk about real roots, we mean that the values of \( x \) are real numbers, not imaginary or complex ones. In many practical applications, such as physics or engineering, real roots represent actual quantities you can measure, like length or time.

• The quadratic formula helps find these real roots when the discriminant is zero or positive.
• In our example, we found the roots to be \( x = 6 \) and \( x = -10 \).

These values satisfy the original equation \( x^2 + 4x - 60 = 0 \), meaning if you substitute them back into the equation, both will hold true. Real roots are key in understanding the solutions to the problems described by quadratic equations.

Solving Quadratics

Solving quadratic equations involves finding the roots of the equation. You have several methods to choose from, including factoring, completing the square, graphing, or using the quadratic formula. Each method has its own strengths and is suitable for different kinds of problems.

• The quadratic formula is a general method that works for any kind of quadratic equation.
• When the equation can be easily factored, that method is usually quicker.
• Completing the square is ideal for getting the equation into vertex form, which can be helpful in graphing.

In our problem, we used the quadratic formula because it efficiently provided the solution, especially since the discriminant was straightforward to calculate. By following a step-by-step approach, you can solve any quadratic equation and understand the significance of the resulting roots.