Problem 44 - Algebra 2

Question

Solve each equation using any method. When necessary, round real solutions to the nearest hundredth. For imaginary solutions, write exact solutions. $$ 2 x^{2}+4 x=10 $$

Step-by-Step Solution

Answer

The solutions to the equation are approximately \(x \approx -3.45\) and \(x \approx 1.45\), rounded to the nearest hundredth.

Step 1: Rewrite the equation in standard form

To solve the quadratic equation, first rewrite it in standard form, which is \(ax^2 + bx + c = 0\). Subtract 10 from both sides to achieve this form: \(2x^2 + 4x - 10 = 0\).

Step 2: Solve the quadratic equation

Apply the quadratic formula \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\) to find the solutions of the equation. Here, \(a = 2\), \(b = 4\), and \(c = -10\). Insert these values into the formula and simplify to find the values of \(x\).

Step 3: Calculate the discriminant

Before solving, calculate the discriminant \(D = b^2 - 4ac\) to determine the nature of the roots. For our equation, \(D = 4^2 - 4 \cdot 2 \cdot (-10) = 16 + 80 = 96\), which is positive, indicating that there are two real and distinct solutions.

Step 4: Find the solutions

Use the quadratic formula with the calculated values: \(x = \frac{{-4 \pm \sqrt{96}}}{{4}}\). Simplify the square root \(\sqrt{96}\) to \(4\sqrt{6}\) and simplify the formula to get: \(x = \frac{{-4 \pm 4\sqrt{6}}}{{4}}\), which simplifies to \(x = -1 \pm \sqrt{6}\). The solutions are \(x \approx -3.45\) and \(x \approx 1.45\), rounded to the nearest hundredth.

Key Concepts

Quadratic Formula

Solving quadratic equations can be made systematic and straightforward by using the quadratic formula. This powerful tool is expressed as
\[\begin{equation} x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\end{equation}\]
where the equation is in standard form. When you encounter a quadratic equation like
\[\begin{equation} 2x^2 + 4x - 10 = 0\end{equation}\],
identifying coefficients for a (the quadratic coefficient), b (the linear coefficient), and c (the constant term) is essential. In the provided exercise, a = 2, b = 4, and c = -10. These values can be plugged into the quadratic formula.

Using this method allows for the computation of the possible values for x that will satisfy the quadratic equation. It is imperative to observe the \pm sign in the formula, which indicates that there are generally two solutions. The quadratic formula is the most efficient method for solving any quadratic equation, whether it yields real or complex roots.

Discriminant

A crucial step in determining the nature of the roots of a quadratic equation is calculating the discriminant, often denoted as D. The discriminant provides valuable insights into the characteristics of the roots without needing to compute them. It is defined as
\[\begin{equation} D = b^2 - 4ac\end{equation}\]
In our example,
\[\begin{equation} D = 4^2 - 4 \times 2 \times (-10)\end{equation}\]
The discriminant can yield three different scenarios:

• If D > 0, there are two distinct real roots.
• If D = 0, there is exactly one real root (also known as a repeated or double root).
• If D < 0, the roots are complex and come in a conjugate pair.

In the provided exercise, the discriminant D is 96, which is greater than zero, signifying that we have two distinct real roots for the quadratic equation. It is important to note that the discriminant does not provide the roots themselves, but it does give useful information about their nature, informing whether to expect real numbers or if complex numbers might come into play.

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is essential to the process of solving using the quadratic formula and determining the discriminant. The general expression for the standard form is
\[\begin{equation} ax^2 + bx + c = 0\end{equation}\]
The equation 2x^2 + 4x = 10 from our exercise requires manipulation to fit this standard form. This is achieved by moving all terms to one side, resulting in
\[\begin{equation} 2x^2 + 4x - 10 = 0\end{equation}\],
which aligns with the shape of the standard form, where a, b, and c represent real numbers, with a eq 0. The leading coefficient a is particularly significant since if a were zero, the equation would linear rather than quadratic. By having the equation in standard form, you can now identify the coefficients needed for the quadratic formula and proceed with the steps to find the solutions, as demonstrated in the solution steps provided.