Problem 69 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ x^{2}-16 x+64=0 $$

Step-by-Step Solution

Answer

The equation has one real double root: \( x = 8 \).

Step 1: Identify the Equation Type

The given equation is a quadratic equation of the form \( ax^2 + bx + c = 0 \). In this case, \( a = 1 \), \( b = -16 \), and \( c = 64 \).

Step 2: Apply the Quadratic Formula

The quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) is used to find the solutions of the quadratic equation. Here, we will apply this formula to get the roots of the equation.

Step 3: Calculate the Discriminant

The discriminant of the equation is given by \( b^2 - 4ac \). Calculate the discriminant: \((-16)^2 - 4(1)(64) = 256 - 256 = 0\). A discriminant of 0 indicates one real double root.

Step 4: Solve for the Root

Since the discriminant is 0, there is one solution given by \( x = \frac{-b}{2a} \). Substitute the values into the formula: \( x = \frac{-(-16)}{2(1)} = \frac{16}{2} = 8 \).

Step 5: Verify the Solution

Verify by substituting \( x = 8 \) back into the original equation: \( 8^2 - 16 \times 8 + 64 = 64 - 128 + 64 = 0 \), confirming it satisfies the equation.

Key Concepts

Quadratic Formula

In the world of mathematics, particularly when dealing with quadratic equations, the quadratic formula is like a magical key that unlocks the secrets to finding the roots of the equation. A quadratic equation is any equation of the form \( ax^2 + bx + c = 0 \). The quadratic formula provides a straightforward way to determine the solutions for these equations. It states:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula helps us calculate the values of \( x \) that satisfy the equation. To use the quadratic formula, you need to know the coefficients \( a \), \( b \), and \( c \), which are taken directly from the equation.

• \( a \) is the coefficient of \( x^2 \)
• \( b \) is the coefficient of \( x \)
• \( c \) is the constant term

In our example, where the equation is \( x^2 - 16x + 64 = 0 \), the coefficients are \( a = 1 \), \( b = -16 \), and \( c = 64 \). After you plug these values into the formula, you can solve for \( x \) to find the roots of the equation.

Discriminant

The discriminant is a crucial part of the quadratic formula, often represented by \( \Delta \), and given by the expression \( b^2 - 4ac \). The value of the discriminant helps us determine the nature of the roots of a quadratic equation without actually solving it completely.There are three key scenarios:

• If \( \Delta > 0 \), the equation has two distinct real roots.
• If \( \Delta = 0 \), there is exactly one real root, meaning the roots are real and repeated.
• If \( \Delta < 0 \), the equation has two complex roots, meaning no real roots exist.

For the equation \( x^2 - 16x + 64 = 0 \), the discriminant is calculated as:\[(-16)^2 - 4 \cdot 1 \cdot 64 = 256 - 256 = 0\]Since the discriminant is zero, this indicates there is exactly one real root for the given equation.

Real Roots

Real roots are the solutions to a quadratic equation where the discriminant is greater than or equal to zero, meaning they can be plotted on the real number line.

• If there are two real roots, they are generally distinct numbers that satisfy the equation.
• If there is one real root (when the discriminant is zero), it is a repeated root, essentially meaning that both roots are the same.

In our solved example, the equation \( x^2 - 16x + 64 = 0 \) has a discriminant of zero, leading us to find a single real root. By substituting back into the quadratic formula, you simplify it for the case of \( \Delta = 0 \):\[x = \frac{-b}{2a}\]Substituting \( b = -16 \) and \( a = 1 \):\[x = \frac{-(-16)}{2 \cdot 1} = \frac{16}{2} = 8\]Thus, the equation has a real root at \( x = 8 \). This shows once more that understanding the discriminant not only simplifies solving quadratic equations but also helps predict the nature of their solutions.