(Algebra: solve quadratic equations) The two roots of a...

(Algebra: solve quadratic equations) The two roots of a quadratic equation $a x^{2}+b x+c=0$ can be obtained using the following formula: \[r_{1}=\frac{-b+\sqrt{b^{2}-4 a c}}{2 a} \text { and } r_{2}=\frac{-b-\sqrt{b^{2}-4 a c}}{2 a}\] $b^{2}-4 a c$ is called the discriminant of the quadratic equation. If it is positive, the equation has two real roots. If it is zero, the equation has one root. If it is negative, the equation has no real roots. Write a program that prompts the user to enter values for $a, b,$ and $c$ and displays the result based on the discriminant. If the discriminant is positive, display two roots. If the discriminant is $0,$ display one root. Otherwise, display "The equation has no real roots". Note that you can use Math.pow(x, 0.5) to compute $\sqrt{x}$. Here are some sample runs.

(Algebra: solve quadratic equations) The two roots of a quadratic equation $a x^{2}+b x+c=0$ can be obtained using the following formula:
\[r_{1}=\frac{-b+\sqrt{b^{2}-4 a c}}{2 a} \text { and } r_{2}=\frac{-b-\sqrt{b^{2}-4 a c}}{2 a}\]
$b^{2}-4 a c$ is called the discriminant of the quadratic equation. If it is positive, the equation has two real roots. If it is zero, the equation has one root. If it is negative, the equation has no real roots.
Write a program that prompts the user to enter values for $a, b,$ and $c$ and displays the result based on the discriminant. If the discriminant is positive, display two roots. If the discriminant is $0,$ display one root. Otherwise, display "The equation has no real roots".
Note that you can use Math.pow(x, 0.5) to compute $\sqrt{x}$. Here are some sample runs.

Key Concepts

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0, where a, b, and c are constants and a is non-zero. This type of equation is fundamental in algebra due to its parabolic graph, and solving it is essential in various fields of mathematics and applied sciences.

Quadratic Formula

The quadratic formula provides a method for solving any quadratic equation. It is given by (-b ± ?(b² - 4ac)) / (2a), and it explicitly calculates the roots of the equation. This formula is important because it works regardless of whether the roots are real or complex, thus offering a universal approach to quadratic equations.

Discriminant

The discriminant, expressed as b² - 4ac, is a key component of the quadratic formula that determines the nature of the roots of the quadratic equation. A positive discriminant indicates two distinct real roots, zero indicates a double root (a single real solution), and a negative discriminant signifies that the equation has no real roots, only complex ones.

Conditional Logic in Programming

Conditional logic is used in programming to execute different blocks of code based on certain conditions. In the context of solving quadratic equations, it is used to check the value of the discriminant and decide whether to compute two roots, one root, or display that there are no real roots, thereby implementing the mathematical solution in a practical program.

Transcript

00:01 In this video, we will create a program that solves the quadratic equation by prompting the user to enter values for a, b, and c, displaying the result based on the discriminant.

00:08 If the discriminant is positive, the quadratic equation will be solved and the two roots will be displayed.

00:13 If it is zero, the singular root will be calculated and displayed.

00:17 Otherwise, the discriminant is negative, and we will display a message to the user that there are no real roots.

00:23 I've already defined our class as well as the main method with any it.

00:26 I've also imported java .util .scanner, since we know we'll be taking use.

00:30 Or input.

00:31 So the first thing we can do is go into our main method and create a new scanner object by calling scanner with the argument system .in, which allows keyboard input to be read from inside our java application.

00:55 After this, we'll want to prompt the user to enter values a, b, and c, and save h using the method next double of java .utal .scanner class, which scans the next token of the next token of the input as a double so we can use system .out .println to tell the user to enter a value for a.

01:25 After this we're going to want to declare double a equals input.

01:31 Next double.

01:37 So we can go ahead and copy these two lines since we'll be doing the same thing for b and c.

01:52 So we'll be doing the same thing for b and c.

01:57 We can just switch our a's with b's and c's accordingly.

02:08 And at this point, we'll want to declare variable d and calculate our discriminant, since now we have doubles for a, b, and c from our user.

02:18 We can declare double d equals b times b.

02:26 Right now we're looking at our discriminant equation at the right -hand side of the screen.

02:34 B times b squared minus 4 times a times c.

02:42 So now we have our discriminant.

02:45 At this point, we'll want to check whether it is positive 0 or negative.

02:49 So we're going to use an if statement for this...