Problem 17 - College Algebra with Modeling and Visualization

Question

Solve each equation and inequality. Use set-builder or interval notation to write solution sets to the inequalities. (a) \(-4 x^{2}+12 x-9=0\) (b) \(-4 x^{2}+12 x-9<0\) (c) \(-4 x^{2}+12 x-9>0\)

Step-by-Step Solution

Answer

(a) \(x = \frac{3}{2}\); (b) and (c) \(\emptyset\).

Step 1: Recognize the Quadratic Equation

The equation \(-4x^2 + 12x - 9 = 0\) is a quadratic equation, which can be identified by its general form \(ax^2 + bx + c = 0\). Here, \(a = -4\), \(b = 12\), and \(c = -9\).

Step 2: Factor the Quadratic Equation

Attempt to factor the equation \(-4x^2 + 12x - 9 = 0\). By checking factor pairs, we note the equation can be factored as \(-(2x - 3)^2 = 0\).

Step 3: Solve the Factored Equation

Since the equation is \( -(2x - 3)^2 = 0 \), set the expression inside the square equal to zero: \(2x - 3 = 0\). Solve for \(x\) by adding 3 to both sides and then dividing by 2. Thus, \(x = \frac{3}{2}\).

Step 4: Solution for Part (a)

The solution to the equation \(-4x^2 + 12x - 9 = 0\) is \(x = \frac{3}{2}\).

Step 5: Understand the Inequality \(-4x^2 + 12x - 9 < 0\)

This inequality means we are looking for when the expression \(-4x^2 + 12x - 9\) is less than zero. Recall \( -(2x - 3)^2 < 0 \).

Step 6: Identify Solution for Less Than Inequality

Since \( -(2x - 3)^2 < 0 \) does not hold for any real values of \(x\) (since a square is always non-negative, making \(-(2x - 3)^2 \leq 0\)), there are no \(x\) that satisfy this inequality. Thus, the solution is the empty set: \( \emptyset \).

Step 7: Understand the Inequality \(-4x^2 + 12x - 9 > 0\)

This inequality means we are looking for when the expression \(-4x^2 + 12x - 9\) is greater than zero.

Step 8: Identify Solution for Greater Than Inequality

Since \( -(2x - 3)^2 > 0 \) also cannot hold for any real \(x\) (as \( -(2x - 3)^2 \leq 0\)), there are no real values making this true. Therefore, the inequality is also solved as an empty set: \( \emptyset \).

Key Concepts

Factoring Quadratics

Factoring quadratics is a method used to simplify quadratic equations. A quadratic equation is any equation that can be written in the form \( ax^2 + bx + c = 0 \). The key here is to find two numbers that multiply to give \( ac \) and add up to \( b \). In some cases, the quadratic can be factored into simpler binomial expressions.

For the equation \(-4x^2 + 12x - 9 = 0\), you may first simplify using common factors or rearrangements. In this instance, the equation can be factored as \(-(2x - 3)^2 = 0\), indicating that it is a perfect square. The factorization tells us there is only one unique solution to the equation, \(x = \frac{3}{2}\).

Remember that not all quadratic equations can be neatly factored. For those that don't factor easily, other methods like the quadratic formula or completing the square might be necessary. Factoring is often the simplest algebraic solution method, but requires keen insight into patterns and multiplication forms.

Inequalities

Inequalities involve expressions that use the symbols \(<, >, \leq,\) or \(\geq\) instead of an equal sign, indicating a range of possible solutions. Solving a quadratic inequality, such as \(-4x^2 + 12x - 9 < 0\) or \(-4x^2 + 12x - 9 > 0\), involves finding the set of values for which the inequality holds true.

In our example, the expression \(-(2x - 3)^2\) is considered. Recognize that the square of any real number is always non-negative, meaning it is \(\geq 0\). This tells us \(-(2x - 3)^2 \leq 0\) for all real numbers, but never truly negative or positive. Consequently, there are no solutions for \(-4x^2 + 12x - 9 < 0\) or \(-4x^2 + 12x - 9 > 0\), as thus both inequalities resolve to an empty set: \(\emptyset\).

When tackling inequalities, always consider the nature of the quadratic and its possible values by testing, plugging in points, or visualizing it on a graph to understand the range of results.

Quadratic Solution Methods

Solving quadratic equations can sometimes be straightforward, but it often requires multiple methods to test before reaching a solution. Some common methods include:

• Factoring: Attempt this when a quick factor based on simple multiplication is possible.
• Completing the Square: Transform the quadratic into a perfect square trinomial to simplify solving.
• The Quadratic Formula: Use when other methods are cumbersome or factoring is not feasible. Given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

In practice, choosing the right method depends on the equation's properties. For instance, \(-4x^2 + 12x - 9 = 0\) was easily solved by factoring since it reduced to a perfect square.

If you hit a roadblock with factoring and completing the square seems too complex, the quadratic formula is a reliable fallback method, capable of yielding solutions even for equations that don't neatly factor.