Problem 52 - Algebra 1

Question

Solve the equation. $$2 x^{2}+12 x-6=0$$

Step-by-Step Solution

Answer

The roots of the given quadratic equation are \(x = -12 + \sqrt{192}/4\) and \(x = -12 - \sqrt{192}/4\).

Step 1: Identify the values of a, b, and c

In this equation, the value of a is 2, b is 12, and c is -6.

Step 2: Substitute the values in the quadratic formula

Now, substitute these values in the quadratic formula:\(x = -b \pm \sqrt{b^2 - 4ac}/2a\) which comes to \(x = -12 \pm \sqrt{(12)^2 - 4 * 2 * -6}/2*2

Step 3: Simplify the equation

On simplifying the above equation, the answer comes out to be \(x = -12 \pm \sqrt{144 + 48}/4\) which further simplifies to \(x = -12 \pm \sqrt{192}/4\).

Step 4: Solve the equation

Finally, we get the roots of the quadratic equation as \(x = -12 + \sqrt{192}/4\), and \(x = -12 - \sqrt{192}/4\).

Key Concepts

Quadratic Formula

Understanding the quadratic formula is key to solving quadratic equations. It is a tool that provides a systematic solution for finding the roots of any quadratic equation, which is any equation that can be written in the form \(ax^2 + bx + c = 0\). The general quadratic formula is: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

When you encounter a quadratic equation such as \(2x^2 + 12x - 6 = 0\), the quadratic formula allows you to solve for \(x\) by plugging in the coefficients \(a = 2\), \(b = 12\), and \(c = -6\). It consists of three steps:

• Finding the directive (\(\pm\)), which means you will have two different solutions.
• Calculating the discriminant (\(b^2 - 4ac\)), which tells you the nature of the roots (real and distinct, real and equal, or complex).
• Dividing by \(2a\) to find the actual roots.

Using the formula can seem daunting, but with practice, it becomes a straightforward method for solving quadratics and is especially useful when factoring is difficult or not possible.

Factoring Quadratics

Factoring quadratics is another method to find the roots of a quadratic equation. Unlike using the quadratic formula, factoring involves breaking down the equation into simpler, multipliable units that, when set to zero, reveal the roots.

To factor a quadratic equation, you follow these steps:

• Rewrite the equation in the form \(ax^2 + bx + c = 0\).
• Find two numbers that multiply to \(ac\) and add up to \(b\).
• Rewrite the middle term, \(bx\), using the two numbers you found.
• Group the terms into pairs and factor out the common factor from each pair.
• Use the zero product property, which states that if \(ab = 0\), then either \(a = 0\) or \(b = 0\) (or both), to find the roots.

While factoring is a powerful technique, it's often limited to equations where such pairs of numbers exist and can be somewhat tricky to master. However, when applicable, it's usually quicker and simpler than using the quadratic formula.

Roots of a Quadratic Equation

The roots of a quadratic equation are the points where the graph of the equation intersects the \(x\)-axis. These are the values for \(x\) that make the equation \(ax^2 + bx + c = 0\) true. Roots can either be real or complex and may be distinct or equal depending on the discriminant \(b^2 - 4ac\).

For the example \(2x^2 + 12x - 6 = 0\), by applying the quadratic formula, you'll find the exact points where the parabola touches the \(x\)-axis. Here's how the discriminant affects the nature of the roots:

• If \(b^2-4ac > 0\), the equation has two real and distinct roots.
• If \(b^2-4ac = 0\), the equation has one real root (also called a repeated or double root).
• If \(b^2-4ac < 0\), the equation has two complex roots.

The discriminant in this equation is positive (\(144 + 48\)), indicating two real and distinct roots. The central concept of the roots is that they are the solutions to the equation — the \(x\)-values that satisfy the original quadratic equation. Finding these roots is a fundamental aspect of solving quadratics and can offer insight into the graph and behavior of the quadratic function.