Problem 22 - Algebra for College Students

Question

Solve each equation by using the quadratic formula. $$p^{2}+6 p+4=0$$

Step-by-Step Solution

Answer

The solutions are \[-3 + \sqrt{5}\] and \[-3 - \sqrt{5}\].

Step 1: Identify the coefficients

The equation is in the form of \(ax^{2} + bx + c = 0\). Identify the coefficients: a = 1, b = 6, c = 4.

Step 2: Write the quadratic formula

The quadratic formula to solve \(ax^{2} + bx + c = 0\) is given by \[-\frac{b \pm \sqrt{b^{2} - 4ac}}{2a}\].

Step 3: Substitute the coefficients

Substitute a = 1, b = 6, and c = 4 into the quadratic formula: \[-\frac{6 \pm \sqrt{6^{2} - 4(1)(4)}}{2(1)}\].

Step 4: Simplify inside the square root

Calculate the value inside the square root: \(6^{2} - 4 \cdot 1 \cdot 4 = 36 - 16 = 20\).

Step 5: Simplify the expression

Now substitute back the simplified term: \[-\frac{6 \pm \sqrt{20}}{2} = -\frac{6 \pm 2\sqrt{5}}{2}\].

Step 6: Divide by 2

Finally, divide each term by 2: \[-\frac{6}{2} \pm \frac{2\sqrt{5}}{2} = -3 \pm \sqrt{5}\].

Key Concepts

solving quadratic equations

Quadratic equations are equations of the form \( ax^2 + bx + c = 0 \). The quadratic formula is a powerful tool to solve such equations. It works for any quadratic equation, as long as you correctly identify the coefficients. The formula is \(\frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\). Let's explore how to apply this formula step by step, using our given equation \(p^2 + 6p + 4 = 0\).

step-by-step algebra

Using the quadratic formula involves several key algebraic steps:

Step 1: Identify the coefficients.
We start with the standard quadratic equation form \( ax^2 + bx + c = 0 \). Here, \(a\) is the coefficient of \( x^2 \), \( b \) is the coefficient of \( x \), and \( c \) is the constant term. From our given equation, \(a = 1\), \(b = 6\), and \(c = 4\).

Step 2: Write down the quadratic formula.
The quadratic formula is \( \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \).

Step 3: Substitute the values for \(a\), \(b\), and \(c\).
Replacing the coefficients in the formula, we get: \( \frac{{-6 \pm \sqrt{{36 - 16}}}}{2 \cdot 1} \).

Step 4: Simplify inside the square root.
Calculate the expression inside the square root: \( 36 - 16 = 20 \).

Step 5: Substitute back the simplified square root term.
This gives us: \( \frac{{-6 \pm \sqrt{20}}}{2} = \frac{{-6 \pm 2\sqrt{5}}}{2} \).

Step 6: Divide each term by 2.
Finally, simplifying further, we get: \( -3 \pm \sqrt{5} \). So, the solutions are: \( -3 + \sqrt{5} \) and \( -3 - \sqrt{5} \).

coefficients identification

Identifying the coefficients in a quadratic equation is the first and most crucial step in applying the quadratic formula.

• \text{Coefficient } a: This is the number before the \( x^2 \) term. For our equation \( p^2 + 6p + 4 = 0 \), the coefficient \(a\) is \(1\).
• \text{Coefficient } b: This is the number before the \( x \) term. In our example, \(b\) is \(6\).
• \text{Term } c: This is the constant term without an \( x \) attached to it. Here, \( c \) is \( 4 \).

Once you correctly identify these coefficients, you can plug them into the quadratic formula to find the roots of the equation. Always double-check your identification of coefficients to avoid mistakes in the calculations.