Question

Challenge Problems $$3.22 x^{2}+9.66 x+2.85=0$$

Short answer

The roots of the quadratic equation \(3.22x^2 + 9.66x + 2.85 = 0\) are \(x_1\) and \(x_2\), which are found by using the quadratic formula.

Step-by-step solution

Identify the coefficients

Recognise the quadratic equation in the form of \(ax^2 + bx + c = 0\). Here, \(a = 3.22\), \(b = 9.66\), and \(c = 2.85\).

Apply the quadratic formula

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the roots of the quadratic equation. Replace \(a\), \(b\), and \(c\) with their respective values.

Calculate the discriminant

First, find the discriminant \(\Delta = b^2 - 4ac\) which is \(\Delta = 9.66^2 - 4 \times 3.22 \times 2.85\). Calculate the value of the discriminant.

Evaluate the square root of the discriminant

Compute the square root of the discriminant to use in the quadratic formula.

Compute the roots

Calculate the two possible values for \(x\) by separately evaluating the plus and minus in the quadratic formula. \(x_1 = \frac{-9.66 + \sqrt{\Delta}}{2 \times 3.22}\) and \(x_2 = \frac{-9.66 - \sqrt{\Delta}}{2 \times 3.22}\).

Simplify the solutions

Simplify the fractions to find the roots.

Key concepts

Quadratic Formula

The quadratic formula is a pivotal tool in algebra for solving quadratic equations, which are equations in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients and \(a \eq 0\). The formula is expressed as \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), which provides a method to find the roots of any quadratic equation.

This magical formula derives from the process of completing the square and gives both real and complex solutions. It is an efficient way to ensure no root is missed, as it accounts for both possibilities under the square root sign: the addition and subtraction of the square root of the discriminant (the part under the radical sign). Students can apply this formula directly to any quadratic equation once they have identified the appropriate coefficients \(a\), \(b\), and \(c\), as seen in Step 1 of the exercise solution.

Discriminant

The discriminant in the context of quadratic equations is the part of the quadratic formula under the square root sign, denoted as \(\Delta = b^2 - 4ac\). It plays a crucial role in determining the nature and number of roots for the equation.

When \(\Delta > 0\), the quadratic equation has two distinct real roots. In case \(\Delta = 0\), there is one real root (also called a repeated or double root). If \(\Delta < 0\), the equation has two complex roots. During Step 3 of the exercise solution, the discriminant is calculated, which informs us about the types of roots we can expect before even calculating them. This step is important as it sets the stage for what comes next and helps us anticipate if we should expect real-number solutions or complex-number solutions.

Roots of a Quadratic

The roots of a quadratic equation are the values of \(x\) that make the equation \(ax^2 + bx + c = 0\) true. These roots can also be referred to as the solutions or zeros of the equation. There can be two roots, one root, or no real roots, depending on the value of the discriminant.

Using the quadratic formula provides the most comprehensive approach to finding these roots. As demonstrated in Steps 2 to 5, once the coefficients are known and the discriminant calculated, the roots can be found by applying the plus and minus versions of the formula. It is important to compute both roots even if they present complex numbers, which are just as valid in the scope of mathematics.

Solving Quadratic Equations

Solving quadratic equations involves finding the values of \(x\) for which the quadratic equation is satisfied. This process typically requires identifying the coefficients, calculating the discriminant, and then applying the quadratic formula as seen in the provided step-by-step solution. It starts with writing the equation in standard form and proceeds with the systematic application of algebraic methods.

While the quadratic formula is a universally applicable method, alternatives like factoring (when possible), completing the square, or using a graph of the quadratic function to find its zeroes, might also be used depending on the specific equation's characteristics. However, the quadratic formula remains the fail-safe method, especially when dealing with complex coefficients, as it always provides a direct route to the equation's roots, which are the ultimate goal of the solving process.