Problem 40 - Algebra 2

Question

Solve each equation by using the method of your choice. Find exact solutions. \(-4(x+3)^{2}=28\)

Step-by-Step Solution

Answer

The solutions are \(x = -3 + \sqrt{7}i\) and \(x = -3 - \sqrt{7}i\).

Step 1: Simplify the Equation

Start by dividing both sides of the equation by -4 to isolate the squared term. \[ (x+3)^{2} = -7 \]

Step 2: Analyze the Equation

Notice that \((x+3)^{2} = -7\) implies that we need to take the square root of a negative number, which means the solutions are complex numbers.

Step 3: Solve for x

To solve \((x+3)^{2} = -7\), take the square root of both sides:\[ x+3 = \pm \sqrt{-7} \] This gives us:\[ x+3 = \pm \sqrt{7}i \]

Step 4: Isolate x

Subtract 3 from both sides to find \(x\):\[ x = -3 \pm \sqrt{7}i \] These are the exact solutions to the equation.

Key Concepts

Quadratic Equations

In mathematics, quadratic equations are a type of polynomial equation that have a specific form: \[ ax^2 + bx + c = 0 \] where \(a\), \(b\), and \(c\) are constants, and \(a\) must not be zero. Quadratic equations can often be solved by factoring, completing the square, using the quadratic formula, or graphing. They have a wide array of applications in fields ranging from physics to finance. In the exercise, the equation is transformed into a quadratic form as \((x+3)^{2} = -7\) after simplification. Key characteristics of quadratic equations:

• They can have up to two real or complex solutions, based on the discriminant \(b^2 - 4ac\).
• Quadratic equations form a parabola when graphed, and the solutions are the points where the parabola crosses the x-axis.

Understanding these characteristics can help solve quadratic equations more efficiently, especially when they lead to complex solutions.

Imaginary Numbers

Imaginary numbers were introduced when mathematicians tried to solve equations with negative square roots. The basic unit of imaginary numbers is \(i\), defined as \(\sqrt{-1}\). When you take the square root of a negative number, it results in an imaginary number. This is seen in the original problem's solution where after simplifying the equation to \((x+3)^2 = -7\), we proceed with finding that \(\sqrt{-7} = \sqrt{7}i\). Some important aspects of imaginary numbers include:

• They form the basis for complex numbers, where a complex number is expressed as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.
• They are crucial in fields like electrical engineering and physics where they model real-world phenomena.

By grasping the concept of imaginary numbers, you can solve equations that appear "unsolvable" with real numbers.

Complex Solutions

When solving certain quadratic equations, as in our initial problem, the solutions may not be real numbers. Instead, they could be complex numbers. A complex number has both a real part and an imaginary part, generally written as \(a + bi\). In the solution to the given problem, the isolation of \(x\) leads to \(x = -3 \pm \sqrt{7}i\), illustrating two complex solutions. These arise because the square root of a negative number yields an imaginary number.Here are some insights about complex solutions:

• Complex solutions to equations often come in conjugate pairs, like \(-3 + \sqrt{7}i\) and \(-3 - \sqrt{7}i\) in our example.
• Complex solutions are symmetrical with respect to the real axis in the complex plane.

Understanding the nature of complex solutions helps when dealing with quadratic equations, indicating the presence of imaginary components and offering a complete set of solutions to these equations.