Problem 58 - College Algebra and Calculus: An Applied Approach

Question

Solve the quadratic equation using any convenient method. \(9 x^{2}+12 x+3=0\)

Step-by-Step Solution

Answer

The solutions of the given quadratic equation are \(x=-\frac{1}{3}\) and \(x=-1\)

Step 1: Identify coefficients

Firstly, note down the coefficients 'a', 'b', and 'c' from the quadratic equation. Here, \(a=9\), \(b=12\), and \(c=3\)

Step 2: Apply the quadratic formula

Apply the quadratic formula, which is \(-b \pm \sqrt{b^{2}-4ac} \over 2a\). Our 'a' is 9, 'b' is 12, and 'c' is 3. Substituting these values, we get \(-12 \pm \sqrt{(12)^{2}-4*9*3} \over 2*9= -12 \pm \sqrt{144-108} \over 18= -12 \pm \sqrt{36} \over 18\)

Step 3: Solve the roots

After simplifying the above equation, the roots of the equation are \(-12 \pm 6 \over 18\). Which gives us two roots; \(\frac{-12+6}{18}= -\frac{1}{3}\) and \(\frac{-12-6}{18}= -1\)

Key Concepts

Quadratic Formula

The Quadratic Formula is an essential tool when solving quadratic equations. A quadratic equation generally takes the form \(ax^2 + bx + c = 0\). The formula is represented as:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Here's how it helps:

• \(-b\): This part of the formula changes the sign of the 'b' coefficient in the equation.
• \(\pm\): It means two possible solutions, one with a plus and another with a minus.
• \(\sqrt{b^2 - 4ac}\): Known as the discriminant, it indicates the nature of the roots. If this is positive, there are two real roots; if zero, there is one real root; if negative, there are two complex roots.
• \(/ 2a\): Divides everything by twice the coefficient of \(x^2\).

Using this formula can efficiently determine the roots of any quadratic equation. It simplifies what might otherwise be a longer factoring process.

Roots of Quadratic Equations

Finding the roots of quadratic equations means solving for the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). These values of \(x\) are called the roots, and they tell us where the parabola represented by the quadratic equation intersects the x-axis.

When you use the Quadratic Formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), the solutions you find are the roots. The terms inside the formula—particularly the discriminant \(b^2 - 4ac\)—help us understand the type of roots:

• If \(b^2 - 4ac > 0\), there are two distinct real roots, indicating two intercepts on the x-axis.
• If \(b^2 - 4ac = 0\), there is one unique real root, showing a single point of tangency on the x-axis.
• If \(b^2 - 4ac < 0\), the roots are complex, meaning there is no real x-axis intercept.

These roots give us important geometric and algebraic information about the quadratic function.

Solving Quadratics Step by Step

Solving quadratics step by step involves identifying the coefficients, applying the formula, and simplifying the results.

### Step 1: Identify CoefficientsBegin by extracting the values of \(a\), \(b\), and \(c\) from the standard quadratic equation \(ax^2 + bx + c = 0\). These coefficients are crucial as they plug into the Quadratic Formula.

### Step 2: Apply the Quadratic FormulaInsert the coefficients into the formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This step is mostly about substitution and requires careful arithmetic.

### Step 3: Simplify the ResultsSolve the resulting expression from the formula. Look closely at the discriminant to determine the nature of the roots:

• Perform operations like addition or subtraction inside the square root first, then take the square root.
• Complete the arithmetic with the fractions afterward.

Following these steps ensures a systematic approach to solving any quadratic equation.