Problem 19 - Algebra for College Students

Question

Solve each equation by using the quadratic formula. $$9+24 x+16 x^{2}=0$$

Step-by-Step Solution

Answer

x = -3/4

Step 1: Identify coefficients

The general form of a quadratic equation is \[ ax^2 + bx + c = 0 \] In this case, the given equation is \[ 9 + 24x + 16x^2 = 0 \] Rewriting it to match the general form, it becomes: \[ 16x^2 + 24x + 9 = 0 \] Now, identify the coefficients: \( a = 16 \), \( b = 24 \), \( c = 9 \).

Step 2: Write down the quadratic formula

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

Step 3: Substitute the coefficients into the formula

Substitute the values of \( a \), \( b \), and \( c \) into the quadratic formula: \[ x = \frac{-(24) \pm \sqrt{(24)^2 - 4(16)(9)}}{2(16)} \] Simplify the expression inside the square root: \[ (24)^2 = 576 \] \[ 4 \cdot 16 \cdot 9 = 576 \] Therefore: \[ x = \frac{-24 \pm \sqrt{576 - 576}}{32} \] \[ x = \frac{-24 \pm \sqrt{0}}{32} \]

Step 4: Solve for x

Since \( \sqrt{0} = 0 \), the equation simplifies to: \[ x = \frac{-24}{32} \] Reducing the fraction: \[ x = \frac{-3}{4} \]

Key Concepts

Solving Quadratic Equations

A quadratic equation is an equation of the form \[ ax^2 + bx + c = 0 \]. It can often be solved using the quadratic formula, which is useful for equations that are difficult to factor. The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].
This formula will deliver two possible values for x, corresponding to the '+ or -' part in the formula. One solution uses the plus sign and the other uses the minus sign.
This method works with any quadratic equation, but you must always correctly identify the coefficients and substitute them accurately into the formula.

Identifying Coefficients

In a quadratic equation like \[ ax^2 + bx + c = 0 \], the letters a, b, and c represent specific numbers called coefficients. Here's how to identify them in the equation:
1. **a**: This is the coefficient of \( x^2 \). It is the number in front of \( x^2 \).
2. **b**: This is the coefficient of x. It is the number in front of the x term.
3. **c**: This is the constant term without an x.
In the example equation \[ 9 + 24x + 16x^2 = 0 \], we rewrite it to match the general form: \[ 16x^2 + 24x + 9 = 0 \]. Now we see that a = 16, b = 24, and c = 9. Identifying these correctly is crucial for using the quadratic formula.

Simplifying Expressions

After identifying the coefficients and substituting them into the quadratic formula, it's important to simplify the expressions step by step to avoid mistakes. Let's see how it works with
\[ x = \frac{-(24) \pm \sqrt{(24)^2 - 4(16)(9)}}{2(16)} \]. Start by simplifying inside the square root:
\[ (24)^2 = 576 \] and \[ 4 \cdot 16 \cdot 9 = 576 \].
Then, we get
\[ x = \frac{-24 \pm \sqrt{576 - 576}}{32} = \frac{-24 \pm \sqrt{0}}{32} \].
The square root of 0 is 0, so it further simplifies to:
\[ x = \frac{-24}{32} = \frac{-3}{4} \].
Make sure to perform these calculations step by step to avoid errors and arrive at the correct solution.