Problem 38 - Algebra 1

Question

Decide how many solutions the equation has. $$x^{2}+16 x+64=0$$

Step-by-Step Solution

Answer

The quadratic equation has exactly one real solution. This is derived from the fact that the value of the discriminant equals to 0.

Step 1: Identify coefficients

In the quadratic equation \(x^2 + 16x + 64 = 0\), the coefficients of x^2, x, and the constant term are a = 1, b = 16, and c = 64 respectively. We identify these coefficients to apply in the discriminant formula.

Step 2: Apply the Discriminant Formula

Apply the coefficients into the discriminant formula \(D = b^2 - 4ac\). So D equals to \( (16)^2 - 4*1*64 \). Calculate the new value of D.

Step 3: Calculate the Discriminant and Determine the Number of Solutions

D equals to 256 - 256 which equals to 0. Given the value of the discriminant, if D > 0, there are two distinct solutions, if D = 0, there is one real solution, and if D < 0, there aren't any real solutions. Thus, in our case, as the discriminant equals to 0, there is exactly one real solution to this quadratic equation.

Key Concepts

Discriminant

In quadratic equations, the discriminant is a key component in understanding the nature of the roots. It is calculated using the formula \[D = b^2 - 4ac\]. Here, \(b\), \(a\), and \(c\) are the coefficients from the quadratic equation \(ax^2 + bx + c = 0\). This value assists in determining if the roots are real or complex, and whether they are distinct or repeated.

For a quick check on the number of solutions:

• If \(D > 0\), the equation has two distinct real solutions.
• If \(D = 0\), there is exactly one real solution, often called a repeated root.
• If \(D < 0\), the equation has no real solutions, but two complex ones.

By evaluating the discriminant, you get an insightful preview of the solution landscape without solving it completely.

Real Solutions

Real solutions in the context of quadratic equations refer to the values of \(x\) that satisfy the equation \[ax^2 + bx + c = 0\]. These are the points where the parabola represented by the equation intersects the x-axis on a graph. Determining the number of real solutions gives a clearer picture of how this interaction occurs.

When using the discriminant:

• A discriminant of zero indicates that the parabola just touches the x-axis at a single point. This is known as a tangent, with one real root.
• A positive discriminant shows that the parabola crosses the x-axis at two distinct points, representing two different real solutions.
The nature of the real solutions provides insight into the behavior of the quadratic function, highlighting whether solutions are closely linked or distinctly apart.

Coefficients

The coefficients in a quadratic equation \(ax^2 + bx + c = 0\) are numerical or constant factors that provide crucial information about the equation.

These coefficients are defined as:

• \(a\): the leading coefficient, affecting the parabola’s width and direction. If \(a\) is positive, the parabola opens upwards; if negative, it opens downwards.
• \(b\): the coefficient linked with \(x\), influencing the vertex’s position relative to the y-axis.
• \(c\): the constant term, representing the y-intercept of the parabola.

Identifying and analyzing these coefficients is the first essential step in solving quadratic equations, as they provide the necessary inputs for the discriminant and other solution-solving processes.