Problem 57 - Precalculus Student Solutions Manual 5th

Question

Solve each equation using the quadratic formula. $$-4 x^{2}=-12 x+11$$

Step-by-Step Solution

Answer

The solutions are \[x = \frac{3}{2} + \frac{-i\sqrt{2}}{2}\] and \[x = \frac{3}{2} - \frac{-i\sqrt{2}}{2}\].

Step 1 - Rewrite the equation in standard form

Move all terms to one side of the equation to set it to zero:\[-4x^2 + 12x - 11 = 0\]

Step 2 - Identify coefficients

From the quadratic equation \[ax^2 + bx + c = 0\], identify the coefficients: \[a = -4\], \[b = 12\], \[c = -11\]

Step 3 - Write down the quadratic formula

The quadratic formula is: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

Step 4 - Substitute the coefficients into the formula

Substitute \[a = -4\], \[b = 12\], \[c = -11\] into the quadratic formula: \[x = \frac{-12 \pm \sqrt{12^2 - 4(-4)(-11)}}{2(-4)}\]

Step 5 - Simplify under the square root

Calculate the discriminant: \[12^2 - 4(-4)(-11) = 144 - 176 = -32\]Since the discriminant is negative, we have: \[\sqrt{-32} = 4i\sqrt{2}\]

Step 6 - Solve for x using the quadratic formula

Plug the discriminant back into the formula: \[x = \frac{-12 \pm 4i\sqrt{2}}{-8}\]Simplify the fraction: \[x = \frac{-12}{-8} \pm \frac{4i\sqrt{2}}{-8}\]\[x = \frac{3}{2} \pm \frac{-i\sqrt{2}}{2}\]

Step 7 - Present the solution

The solutions to the quadratic equation are: \[x = \frac{3}{2} + \frac{-i\sqrt{2}}{2}\] and \[x = \frac{3}{2} - \frac{-i\sqrt{2}}{2}\]

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of degree 2. They take the form \[ax^2 + bx + c = 0\], where \(a, b,\) and \(c\) are constants and \(a eq 0\). Quadratic equations can have up to two solutions, which might be real or complex numbers.
To solve a quadratic equation, you might use techniques like factoring, completing the square, or the quadratic formula. The quadratic formula is particularly useful when the equation does not factor neatly or when it involves complex solutions (those involving the imaginary unit \(i\)).
Here is how the quadratic formula looks:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].

The formula involves several key steps:

• Identifying the coefficients \(a, b,\) and \(c\).
• Plugging these values into the formula.
• Calculating the discriminant \(b^2 - 4ac\) to determine the nature of the solutions.
• Simplifying to find the values of \(x\).

Quadratic equations open the door to further exploration of algebra and complex numbers, so grasping the fundamental concepts is essential in mathematics.

Complex Solutions

When solving quadratic equations, sometimes you'll encounter a negative discriminant. This means the equation has no real solutions but instead has complex solutions involving the imaginary unit \(i\). The imaginary unit \(i\) is defined as \(i = \sqrt{-1}\).
Complex solutions come in conjugate pairs. If one solution is \(x = a + bi\), then the other solution is \(x = a - bi\).
Let's break down the step in our example where complex numbers appear:
We had \(12^2 - 4(-4)(-11) = -32\), which is a negative number. Taking the square root of -32, we apply the property of imaginary numbers:

• \(\sqrt{-32} = \sqrt{32} \cdot i = 4i\sqrt{2}\)
• This is simplified and placed back in the quadratic formula to find \(x = \frac{3}{2} \pm \frac{-i\sqrt{2}}{2}\).

This demonstrates how negative discriminants lead to solutions within the realm of complex numbers, highlighting the beauty and depth of quadratic equations in algebra.

Discriminant

The discriminant is a specific part of the quadratic formula. It is found inside the square root: \(b^2 - 4ac\). The value of the discriminant tells us about the nature of the roots of the quadratic equation.
Here's how to interpret the discriminant:

• If \(b^2 - 4ac > 0\), the equation has two distinct real solutions.
• If \(b^2 - 4ac = 0\), the equation has exactly one real solution (a repeated root).
• If \(b^2 - 4ac < 0\), the equation has two complex solutions.

In our example, we calculated the discriminant to be \(-32\), which is less than zero. This let us know that our solutions would be complex. Recognizing the significance of the discriminant is crucial in predicting the type of solutions you'll get with a quadratic equation. With this knowledge, it becomes easier to solve quadratic equations and understand their behavior and solutions deeply.