Question

Find the roots of each quadratic by any of the methods shown in this section. Keep three significant digits. For some, use more than one method and compare results. Implicit Functions. $$33-3 x^{2}=10 x$$

Short answer

The roots of the equation \[-3x^2 - 10x + 33 = 0\] are approximately \[x_1 \approx 2.29\] and \[x_2 \approx -4.79\].

Step-by-step solution

- Rewrite the equation in standard quadratic form

Combine all terms on one side to set the equation to zero. Subtract 10x from both sides to obtain the standard form of a quadratic equation: \[-3x^2 - 10x + 33 = 0\].

- Divide through by -3 to simplify the equation

Divide every term by -3 to make the coefficient of the quadratic term 1 (this is optional but simplifies calculations): \[x^2 + \frac{10}{3}x - 11 = 0\].

- Use the quadratic formula to find the roots

The quadratic formula is \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. For our equation, a = 1, b = +\frac{10}{3}, and c = -11. Substitute these values into the formula to get the roots.

- Calculate the discriminant

Calculate the discriminant (\[\text{discriminant} = b^2 - 4ac\]) which is necessary for the quadratic formula: \[\left(\frac{10}{3}\right)^2 - 4(1)(-11)\].

- Evaluate the roots

Using the calculated discriminant, evaluate the two possible roots using the quadratic formula: \[x = \frac{-\frac{10}{3} \pm \sqrt{\left(\frac{10}{3}\right)^2 - 4(1)(-11)}}{2(1)}\]. Calculate the values with a calculator if necessary to maintain three significant digits.

- Calculate the exact roots

When the calculations are complete, you will get two roots, denoted as \[x_1\] and \[x_2\]. These are the solutions to the quadratic equation.

Key concepts

Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of a quadratic equation, which are the values of the variable that satisfy the equation. It is represented as \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where the letters \(a\), \(b\), and \(c\) are the coefficients of the terms in the standard form of the quadratic equation \(ax^2 + bx + c = 0\).

This formula is derived from completing the square process and provides a straightforward method to determine the roots. In using the quadratic formula, determining the values of \(a\), \(b\), and \(c\) is crucial. Once these values are identified, you can substitute them into the formula to find the roots, which might be real or complex numbers. The quadratic formula can solve any quadratic equation, even if factoring is difficult or impossible.

Solving Quadratic Equations

Solving quadratic equations is a foundational skill in algebra. A quadratic equation can be solved by various methods including factoring, completing the square, graphing, and using the quadratic formula.

The choice of method often depends on the specific form of the quadratic equation and the complexity of its coefficients. For instance, if the quadratic can be easily factored, this method is often the quickest. However, for equations that do not factor neatly, the quadratic formula is the most reliable approach. When graphing, the points where the parabola (the graph of a quadratic equation) intersects the x-axis represent the roots of the equation. In the classroom, multiple methods are often taught to provide students with a toolkit to address a variety of quadratic equations they might encounter.

Discriminant Calculation

The discriminant in a quadratic equation is found within the radical of the quadratic formula, represented by \(b^2 - 4ac\). It is key to determining the nature of the roots without actually calculating them.

The discriminant can yield three types of solutions:

• If the discriminant is positive, the equation has two distinct real roots.
• If the discriminant is zero, the equation has exactly one real root (a repeated root).
• If the discriminant is negative, the equation has two complex roots.

This value informs us not only of the number of roots but also of their nature - whether they are real and distinct, real and coincident, or non-real complex numbers. Understanding how to evaluate the discriminant is an essential step when solving quadratic equations using the quadratic formula.

Implicit Functions

Implicit functions are those where the dependent variable is not isolated on one side of the equation. Rather than being in the form of \(y = f(x)\), they are often in the form of \(F(x, y) = 0\).

Solving a quadratic equation that is presented as an implicit function involves rearranging it into the standard form by combining all the terms on one side of the equality. In the provided exercise, the original equation \(33 - 3x^2 = 10x\) is an example of an implicit function. By rearranging terms, we bring it into an explicit standard quadratic form. This step is pivotal for applying all subsequent methods of solving quadratic equations, such as factoring, completing the square, or using the quadratic formula, as has been done to find its roots.