Problem 40 - Algebra 2

Question

Solve each equation by factoring. \(16 x^{2}-48 x=-27\)

Step-by-Step Solution

Answer

Solutions are \(x = \frac{3}{4}\) and \(x = \frac{9}{4}\).

Step 1: Move all terms to one side

First, let's move all terms to the left side of the equation to set it equal to zero. We have:\[ 16x^2 - 48x + 27 = 0 \]

Step 2: Factor the quadratic equation

Next, we want to factor the quadratic expression. This means finding two binomials that multiply to give us the quadratic equation. Factoring the expression \(16x^2 - 48x + 27\) results in:\[ (4x - 3)(4x - 9) = 0 \]

Step 3: Solve each factor set to zero

Now, solve each factor individually by setting them equal to zero:1. \(4x - 3 = 0\)2. \(4x - 9 = 0\) For \(4x - 3 = 0\), add 3 to both sides giving us \(4x = 3\), then divide both sides by 4 to find \(x = \frac{3}{4}\).For \(4x - 9 = 0\), add 9 to both sides giving us \(4x = 9\), then divide by 4 to find \(x = \frac{9}{4}\).

Step 4: Verify the solutions

Lastly, verify the solutions by substituting them back into the original equation to ensure both sides are equal. Substitute \(x = \frac{3}{4}\) and \(x = \frac{9}{4}\) back into \(16x^2 - 48x = -27\) and confirm the equality is true. This confirms both solutions are correct.

Key Concepts

Factoring Quadratics

Factoring quadratics is a method used to solve quadratic equations by expressing them as a product of simpler binomials. This simplifies the process of finding the values of the variable that satisfy the equation. To factor a quadratic equation, you generally look for two numbers that both add to give the coefficient of the middle term and multiply to give the product of the coefficients of the first and last terms.

In the example given:

• We start with the equation: \[16x^2 - 48x + 27 = 0\]
• The goal is to express it as: \[(ax + b)(cx + d) = 0\]
• Finding that \[(4x - 3)(4x - 9) = 0\], satisfies the equation means, multiplying these binomials gives the original quadratic.

Analyzing common factors and testing combinations of pairs can help, and sometimes trial and error may be needed. A well-factored quadratic makes solving much simpler because each factor can then be set equal to zero.

Quadratic Formula

The quadratic formula is a robust tool for solving any quadratic equation of the form \(ax^2 + bx + c = 0\). Its main appeal is its application to all quadratic equations, not just those that can be factored easily. The formula is given by:\[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\].

• Here, the '±' sign means you'll typically get two solutions from this formula.
• The expression under the square root sign, \(b^2 - 4ac\), is called the discriminant. This determines the nature and number of roots.

Using the quadratic formula, students can directly plug in the coefficients from their equation and solve for \(x\). It's especially useful for equations that do not factor neatly. Though not used in this particular problem, understanding this formula is essential for tackling more complex quadratic equations efficiently.

Verification of Solutions

Verification is an important step in solving equations to ensure the solutions are correct. After finding solutions, you should always check them by substituting back into the original equation. This confirms that the values fulfill the equation properly.

For instance, if we have solutions \(x = \frac{3}{4}\) and \(x = \frac{9}{4}\), they can be verified by plugging them back into the initial equation: \[16x^2 - 48x = -27\]

• Substitute \(x = \frac{3}{4}\): after simplification, both sides of the equation should equal.
• Substitute \(x = \frac{9}{4}\): the equation should also balance, confirming both solutions are correct.

Verification ensures that no mistakes were made in the factoring or solving process. It provides confidence in the solutions derived and confirms the understanding of the problem-solving methods used.