Problem 55 - Algebra 2

Question

Solve each equation by using the method of your choice. Find exact solutions. $$ x^{2}+12 x+32=0 $$

Step-by-Step Solution

Answer

The exact solutions are \( x = -4 \) and \( x = -8 \).

Step 1: Identify the Quadratic Equation

The equation given is a quadratic equation in the form of \( ax^2 + bx + c = 0 \). Here, \( a = 1 \), \( b = 12 \), and \( c = 32 \). Our task is to find the values of \( x \) that satisfy this equation.

Step 2: Choose a Method

We will use the quadratic formula to solve this equation. The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

Step 3: Calculate the Discriminant

Calculate the discriminant \( D = b^2 - 4ac \). Substitute the values of \( a \), \( b \), and \( c \) into the formula: \[ D = 12^2 - 4 \times 1 \times 32 = 144 - 128 = 16.\]

Step 4: Apply the Quadratic Formula

Since the discriminant is positive, there are two real roots. Substitute \( a \), \( b \), and \( D \) into the quadratic formula: \[ x = \frac{-12 \pm \sqrt{16}}{2 \times 1} = \frac{-12 \pm 4}{2}.\]

Step 5: Solve for Roots

Calculate the two possible values of \( x \): - For \( x = \frac{-12 + 4}{2} = \frac{-8}{2} = -4 \).- For \( x = \frac{-12 - 4}{2} = \frac{-16}{2} = -8 \).

Step 6: State the Exact Solutions

The exact solutions of the equation \( x^2 + 12x + 32 = 0 \) are \( x = -4 \) and \( x = -8 \).

Key Concepts

Discriminant

The discriminant is a key part of the quadratic formula and is represented as \( D = b^2 - 4ac \). It helps us to determine the nature of the roots of a quadratic equation. In simpler terms, it's like a guide that tells us what kind of solutions we can expect from our equation.

• If \( D > 0 \), we have two distinct real roots.
• If \( D = 0 \), there is one real root that is repeated (also known as a double root).
• If \( D < 0 \), there are no real roots, but two complex roots.

In our problem, the discriminant \( D = 16 \) is positive, which indicates that the quadratic equation has two distinct real roots. This is an important insight that helps us predict the type of solutions before we even solve the equation.

Quadratic Equation

A quadratic equation is any equation that can be rearranged in standard form as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. The term \( ax^2 \) signifies that the highest power of \( x \) is 2, making it "quadratic." This is what sets quadratic equations apart from linear equations, which only have \( x \) to the power of 1.

In our exercise, \( x^2 + 12x + 32 = 0 \) is the quadratic equation, where \( a = 1 \), \( b = 12 \), and \( c = 32 \). Solving quadratic equations involves finding the value(s) of \( x \) that make the equation true, which can be done by various methods such as factoring, completing the square, or using the quadratic formula. These methods rely on understanding the structure and properties of quadratic equations.

Real Roots

In the context of quadratic equations, real roots are the solutions to the equation where the graph of the quadratic intersects the x-axis. Every quadratic equation potentially has up to two solutions. When the discriminant is positive as in our problem (\( D = 16 \)), the quadratic equation has two distinct real roots.

Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), we find these roots by substituting the known values of \( a \), \( b \), and \( c \). For instance, in the equation \( x^2 + 12x + 32 = 0 \), we substitute these into the formula to find:

• \( x = \frac{-12 + 4}{2} = -4 \)
• \( x = \frac{-12 - 4}{2} = -8 \)

These values, \( x = -4 \) and \( x = -8 \), are the real roots of the equation. Real roots signify actual crossings on the x-axis, providing tangible solutions to the quadratic puzzle.