Problem 21 - Algebra 2 and Trigonometry

Question

In \(9-26,\) solve each quadratic equation by completing the square. Express the answer in simplest radical form. $$ 2 x^{2}+12 x+3=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = -3 \pm \sqrt{\frac{15}{2}}\).

Step 1: Move the constant to the other side

Start by moving the constant term to the right side of the equation.\[ 2x^2 + 12x + 3 = 0 \]Subtract 3 from both sides:\[ 2x^2 + 12x = -3 \]

Step 2: Normalize the quadratic coefficient

Next, divide the entire equation by 2, the coefficient of \(x^2\), to make it 1.\[ x^2 + 6x = -\frac{3}{2} \]

Step 3: Complete the square

To complete the square, take half of the coefficient of \(x\), square it, and add it to both sides.The coefficient of \(x\) is 6, half of it is 3, and its square is 9. Add 9 to both sides:\[ x^2 + 6x + 9 = -\frac{3}{2} + 9 \]

Step 4: Write in perfect square form

The left side is now a perfect square trinomial:\[ (x + 3)^2 = \frac{15}{2} \]

Step 5: Solve for x using the square root

Take the square root of both sides to solve for \(x\).\[ x + 3 = \pm \sqrt{\frac{15}{2}} \]

Step 6: Isolate x

Subtract 3 from both sides to solve for \(x\):\[ x = -3 \pm \sqrt{\frac{15}{2}} \]

Key Concepts

Quadratic Equations

Quadratic equations are fundamental in algebra and take the form: \( ax^2 + bx + c = 0 \). They involve variables raised to the power of two, known as the quadratic term. The 'a', 'b', and 'c' are constants, and the solutions to these equations are where the equation equals zero.
Quadratic equations can represent various real-world scenarios, such as projectile motion or areas of squares and rectangles. Solving them offers insights into diverse mathematical and practical applications.

Solving Equations

There are several methods for solving quadratic equations: factoring, graphing, and using the quadratic formula. Completing the square is another key technique.
This method involves altering a quadratic into a perfect square trinomial, making it easier to solve. Perfect square trinomials allow for straightforward extraction of roots, simplifying the process of finding equation solutions.

Radical Form

Simplest radical form requires expressing numbers under a square root sign in their simplest terms. While solving quadratic equations, it's common to end with roots in radical form especially when exact values are needed.
For instance, when solving \( x = -3 \pm \sqrt{\frac{15}{2}} \), the solutions are expressed in radical form, providing more precise answers than decimal approximations.

Algebra 2

Algebra 2 delves deeper into solving equations, including quadratic equations using methods like completing the square. This course is essential for mastering concepts needed in advanced math topics.

• It introduces students to more complex problem-solving skills.
• Offers a comprehensive understanding of equations and functions.
• Lays the groundwork for analyzing and interpreting data mathematically.

Algebra 2 is crucial for students aiming to enhance their analytical skills.

Trinomials

A trinomial is a polynomial with three terms, typically of the form \( ax^2 + bx + c \). When solving quadratic equations, converting a quadratic expression into a trinomial is useful.
Completing the square manipulates the quadratic into a perfect square trinomial, facilitating easier equation solving. Recognizing such patterns is a vital skill in simplifying complex equations.