Problem 15 - Algebra 2 and Trigonometry

Question

In \(3-14\) , use the quadratic formula to find the roots of each equation. Irrational roots should be written in simplest radical form. $$ x^{2}-6 x+3=0 $$

Step-by-Step Solution

Answer

The roots are \(x = 3 + \sqrt{6}\) and \(x = 3 - \sqrt{6}\).

Step 1: Identify the coefficients

The given quadratic equation is \(x^2 - 6x + 3 = 0\). Identify the coefficients \(a\), \(b\), and \(c\) for the quadratic formula \(ax^2+bx+c=0\). Here, \(a = 1\), \(b = -6\), and \(c = 3\).

Step 2: Write the quadratic formula

The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This formula is used to find the roots of any quadratic equation of the form \(ax^2+bx+c=0\).

Step 3: Calculate the discriminant

The discriminant is the part under the square root in the quadratic formula, \(b^2 - 4ac\). Calculate it using the identified coefficients: \((-6)^2 - 4(1)(3) = 36 - 12 = 24\).

Step 4: Apply the quadratic formula

Substitute the values \(a\), \(b\), and the discriminant into the quadratic formula. This gives: \[x = \frac{-(-6) \pm \sqrt{24}}{2(1)} = \frac{6 \pm \sqrt{24}}{2}\]

Step 5: Simplify the radical and divide

Simplify the irrational part: \(\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}\). Substitute back:\[x = \frac{6 \pm 2\sqrt{6}}{2}\]Simplify by dividing each term by 2:\[x = 3 \pm \sqrt{6}\]

Step 6: Write the roots

Finally, the rational roots of the quadratic equation \(x^2 - 6x + 3 = 0\) are \(x = 3 + \sqrt{6}\) and \(x = 3 - \sqrt{6}\).

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental part of algebra. They represent polynomial equations of degree two and generally have the form \( ax^2 + bx + c = 0 \). This type of equation can graphically represent parabolas, which means they open upwards or downwards on a coordinate plane.
To solve a quadratic equation, several methods can be used, such as factoring, completing the square, or using the quadratic formula. Each method provides a way to find the values of \( x \) that satisfy the equation.

• **Factoring** involves expressing the quadratic as a product of two binomials.
• **Completing the square** transforms the equation so that one side is a perfect square trinomial.
• **The quadratic formula** is a powerful tool that can find roots for any quadratic equation, typically used when other methods are not straightforward.

Finally, it's important to know that a quadratic equation might have:

• two distinct real roots
• one real root (a double root)
• or two complex roots

depending on the value of its **discriminant**.

Discriminant

The discriminant plays a crucial role in understanding the nature of the solutions of a quadratic equation. It is found under the square root symbol in the quadratic formula and is constructed as \( b^2 - 4ac \). This value helps determine how many and what type of roots a quadratic equation has.
When the discriminant is:

• **Positive**: There are two distinct real roots. This means the parabola crosses the x-axis at two points.
• **Zero**: There is one real root. The parabola touches the x-axis at one point. This root is called a double root.
• **Negative**: There are no real roots, but two complex roots. The parabola doesn't intersect the x-axis.

In the solved exercise above, the discriminant is 24, which is positive. Thus, the quadratic equation has two distinct real irrational roots.

Radical Simplification

Radical simplification is about simplifying the square root component in the quadratic formula to its simplest form. To simplify \( \sqrt{24} \), we can break down the number inside the square root into its prime factors.
Here are the steps:

• Identify the perfect squares within 24. We know that \( 4 \) is a perfect square, and it fits into 24 to produce \( 24 = 4 \times 6 \).
• Simplify \( \sqrt{24} \) using this factorization: \( \sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} \).
• Since \( \sqrt{4} = 2 \), the expression simplifies to \( 2\sqrt{6} \).

Simplifying radicals makes it easier to work with equations and helps in presenting cleaner, more simplified solutions. The solved quadratic equation has its roots expressed with the simplification \( x = 3 \pm \sqrt{6} \), showcasing the importance of this step in finding the simplest form of the solution.