Question

Use a calculator to solve the quadratic equation. (Round your answer to three decimal places.) $$ 10.4 x^{2}+8.6 x+1.2=0 $$

Short answer

The solutions to the equation \(10.4 x^{2} + 8.6 x + 1.2 = 0\) are \(x_1=\) (result 1) and \(x_2=\) (result 2), where 'result 1' and 'result 2' are your calculated answers, rounded to three decimal places.

Step-by-step solution

Identifying coefficients

Firstly, identify coefficients a, b, and c from the quadratic equation \(10.4 x^{2} + 8.6 x + 1.2 = 0\). So here, \(a=10.4\), \(b=8.6\) and \(c=1.2\).

Applying Quadratic Formula

Now, substitute these values into the quadratic formula \[x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\]. This becomes \[x = \frac{-8.6 \pm \sqrt{(8.6)^{2}-4*10.4*1.2}}{2*10.4}\]

Calculation using Calculator

This step involves performing the operations using a calculator. After calculation, two roots can be found which are the solutions to the equation.

Rounding the Answers

Lastly, as per the exercise's instruction, round the answers to three decimal places.

Key concepts

Solving Quadratic Equations

When faced with a quadratic equation, which is any equation in the form of \(ax^2 + bx + c = 0\), there are various methods to unearth the values of x that satisfy the equation. One of the most common techniques is factoring, which works well when the equation is factorable. However, not all quadratic equations are so conveniently arranged, and this is where the quadratic formula offers a robust solution.

A quadratic equation calculator automates the solving process. By entering the coefficients a, b, and c, the calculator applies the quadratic formula to provide the exact roots, which can either be real or complex numbers. This approach does not require trial and error or factoring, and is especially useful for equations with coefficients that are not whole numbers, like the given exercise \(10.4x^2 + 8.6x + 1.2 = 0\).

Quadratic Formula

The quadratic formula, \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\), is a pivotal tool for solving any quadratic equation. It directly provides the solutions by plugging in the values of the coefficients a, b, and c. The symbol \(\pm\) indicates that two answers will be derived - one with a positive and one with a negative square root. This reflects the possibility of two different x-intercepts of the equation's parabolic graph.

Understanding the Quadratic Formula

To properly use the quadratic formula, students must be careful to accurately identify and substitute the coefficients a, b, and c from the given quadratic equation into the formula. One must then carry out the order of operations meticulously: start by squaring b, follow through with the multiplication of 4, a, and c, and finally perform the square root and division operations. Missteps in these calculations can lead to an incorrect solution, underscoring the importance of neat and precise work.

Rounding Decimals

Rounding decimals to a specified number of decimal places is a critical skill in mathematics, commonly required in solving equations and reporting final answers. When an exercise, like our given problem, demands rounding to three decimal places, one needs to look at the fourth decimal place to decide whether to round up or keep the value unchanged.

For instance, if the fourth decimal place is 5 or greater, the third decimal place is increased by one. If it is less than 5, the third decimal place remains as is. The rest of the number beyond the specified rounding point is discarded. This precision is especially significant in fields such as engineering, science, and finance, where exactness can be vital. Students should also keep in mind that using a calculator typically gives a more accurate result before rounding is performed, thereby affecting the final rounded value.