Problem 50 - Algebra 1

Question

Write the quadratic equation in standard form. Solve using the quadratic formula. $$-1+3 x^{2}=2 x$$

Step-by-Step Solution

Answer

The quadratic equation does not have a real solution because the discriminant is negative. It instead has two complex solutions given by \(x = \frac{1}{3} \pm \frac{\sqrt{-8}}{6}\).

Step 1: Write the equation in standard form

To have the equation in standard form, we must arrange it as \(ax^{2} + bx + c = 0\). Our equation is \(-1 + 3x^{2} = 2x\), which rearranges to \(3x^{2} - 2x + 1 = 0\). So, \(a = 3\), \(b = -2\), and \(c = 1\).

Step 2: Apply the Quadratic Formula

The quadratic formula is given by \(x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\). Substituting the values we have \(a = 3\), \(b = -2\), and \(c = 1\) into this formula gives \(x = \frac{-(-2) \pm \sqrt{(-2)^{2}-4 * 3* 1}}{2 * 3}\) which simplifies to \(x = \frac{2 \pm \sqrt{4 - 12}}{6}\).

Step 3: Simplify the equation

Continue simplifying to get \(x = \frac{2 \pm \sqrt{-8}}{6}\). As the term inside the square root is negative, this means that the quadratic equation has no real solution. Generally, if the term \(b^{2}-4ac\) known as the discriminant is negative, then the quadratic equation has complex solutions (imaginary).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\). This formula provides the solutions to any quadratic equation, provided the coefficients \(a\), \(b\), and \(c\) are known. The quadratic formula is given by the expression:

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

This formula calculates the values of \(x\) that satisfy the equation by using the coefficients \(a\), \(b\), and \(c\). The use of "plus-minus" (\(\pm\)) in the formula indicates there may be two possible solutions for \(x\).
Once the equation is in standard form, this formula can be used to find out whether the solutions are real or complex, based on the value inside the square root, known as the discriminant.

Standard Form

The term "standard form" in the context of quadratic equations refers to the arrangement of a quadratic equation as \(ax^2 + bx + c = 0\). This format helps in applying the quadratic formula smoothly.
To convert an equation into its standard form, rearrange all terms such that the equation equals zero. Take, for example, the equation \(-1 + 3x^2 = 2x\). To bring it into the standard form, we must rearrange it by moving all terms to one side, resulting in \(3x^2 - 2x + 1 = 0\).
Understanding the standard form helps identify the necessary coefficients \(a\), \(b\), and \(c\) needed for further calculations such as using the quadratic formula.

Discriminant

The discriminant is a key part of the quadratic formula, located under the square root symbol: \(b^2 - 4ac\). It helps determine the nature of the solutions for a quadratic equation:

• If \(b^2 - 4ac > 0\), there are two distinct real solutions.
• If \(b^2 - 4ac = 0\), there is exactly one real solution (a repeated root).
• If \(b^2 - 4ac < 0\), the solutions are complex (involving imaginary numbers).

In our exercise, the discriminant calculation \(4 - 12 = -8\) is negative, indicating that the equation does not have real solutions, but rather complex ones. The discriminant thus provides valuable information on the types of solutions to expect.

Complex Solutions

Complex solutions arise when the discriminant is negative, indicating that the solutions of the quadratic equation will involve imaginary numbers. In mathematics, an imaginary unit \(i\) is used where \(i^2 = -1\).
When you simplify a square root of a negative number, such as \(\sqrt{-8}\), you express it using \(i\). Thus, \(\sqrt{-8} = \sqrt{8}i\).
In our example, substituting this into the equation gives us complex solutions \(x = \frac{2 \pm 2\sqrt{2}i}{6}\), which simplifies further. Understanding complex solutions paves the way for exploring more advanced mathematical concepts involving complex numbers.