Question

Challenge Problems $$5.11 x^{2}+18.6 x+3.88=0$$

Short answer

The solutions to the equation are \(x_1 = \frac{-18.6 + \sqrt{(18.6)^{2}-4(5.11)(3.88)}}{2(5.11)}\) and \(x_2 = \frac{-18.6 - \sqrt{(18.6)^{2}-4(5.11)(3.88)}}{2(5.11)}\).

Step-by-step solution

Identify the coefficients of the quadratic equation

A quadratic equation is generally written as \(ax^{2} + bx + c = 0\). For the given equation \(5.11x^{2}+18.6x+3.88=0\), identify the coefficients as \(a = 5.11\), \(b = 18.6\), and \(c = 3.88\).

Apply the quadratic formula

The solutions to a quadratic equation \(ax^{2} + bx + c = 0\) can be found using the quadratic formula \(x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\). Substitute \(a\), \(b\), and \(c\) into the formula to find the values of \(x\).

Calculate the discriminant

Calculate the discriminant \(\Delta = b^{2}-4ac\) to determine the nature of the roots. In this equation, \(\Delta = (18.6)^{2} - 4(5.11)(3.88)\).

Calculate the roots

Use the discriminant to calculate the two possible values for \(x\) by substituting the values back into the quadratic formula. The roots are \(x_1 = \frac{-b + \sqrt{\Delta}}{2a}\) and \(x_2 = \frac{-b - \sqrt{\Delta}}{2a}\).

Simplify the solutions

Simplify the expressions obtained for \(x_1\) and \(x_2\) to find the exact values of the roots.

Key concepts

Quadratic Formula

The quadratic formula is a powerful tool that allows us to find the solutions to any quadratic equation, which is an equation of the form \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are known as the coefficients, and \(x\) represents the unknown variable we're trying to solve for. The formula itself is \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\).

Using this formula, we can immediately determine the values of \(x\) that make the original equation true. It's handy because it works for any quadratic equation, regardless of the coefficients. To use it effectively, one must input the correct coefficient values and carry out the operations diligently. This includes careful handling of the square root and the plus or minus sign, which indicates that two solutions (roots) are generally possible for a quadratic equation. More on these roots next!

Discriminant Calculation

The discriminant in a quadratic equation, represented by the Greek letter Delta \(\Delta\), is the expression under the square root in the quadratic formula: \(\Delta = b^2 - 4ac\). It plays a critical role in determining the nature of the roots of the quadratic equation. The value of the discriminant can tell us whether the roots are real or complex and whether they are distinct or repeated.

• If \(\Delta > 0\), the equation will have two distinct real roots.
• If \(\Delta = 0\), the equation will have exactly one real root (also called a repeated or double root).
• If \(\Delta < 0\), the equation will have two complex roots.

Understanding the discriminant can thus give us a good sense of what type of solutions we can expect before even solving the equation.

Roots of a Quadratic Equation

The roots of a quadratic equation are the values of \(x\) that satisfy the equation, essentially where the curve intersects the \(x\)-axis on a graph. Depending upon the value of the discriminant (\(\Delta\)), we get different kinds of roots. If the discriminant is positive, we get two real and distinct solutions. With a discriminant of zero, we encounter a perfect square in the quadratic formula, resulting in one real solution that is repeated. Lastly, a negative discriminant leads to complex roots.

By substituting the determined discriminant in the quadratic formula, one can find these roots and thereby solve the quadratic equation completely. Each root, represented as \(x_1\) and \(x_2\) in the formula, is important for a comprehensive understanding of the equation's behavior.

Identifying Coefficients

Coefficient identification is the first crucial step in solving a quadratic equation. Coefficients are the numerical factors of the terms in the equation: \(a\) is the coefficient of \(x^2\), the term with the highest degree; \(b\) is the coefficient of the first-degree term \(x\); and \(c\) is the constant term, or the term without any variable. For example, in the equation \(5.11x^2 + 18.6x + 3.88 = 0\), the values of the coefficients are \(a = 5.11\), \(b = 18.6\), and \(c = 3.88\).

Correctly identifying these values is imperative because an error in this initial step can lead to incorrect roots. When we look at a quadratic equation, recognizing and accurately identifying each coefficient allows us to utilize the quadratic formula effectively and confidently solve for \(x\).