Solve The Quadratic Equation $x^2 - 20x + 100 = 0$.Step 2 Of 2: Use The Discriminant, $b^2 - 4ac$, To Determine The Number Of Solutions Of The Given Quadratic Equation. Then Solve The Quadratic Equation Using The Formula $x =

Feb 26, 2025 by ADMIN 226 views

**Solve the Quadratic Equation: A Step-by-Step Guide**

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will guide you through the process of solving a quadratic equation using the discriminant and the quadratic formula. We will use the equation $x^2 - 20x + 100 = 0$ as an example.

Step 1: Understand the Quadratic Equation

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants. In our example, the equation is $x^2 - 20x + 100 = 0$ , where $a = 1$ , $b = -20$ , and $c = 100$ .

Step 2: Use the Discriminant to Determine the Number of Solutions

The discriminant is a value that can be calculated from the coefficients of the quadratic equation. It is denoted by the symbol $\Delta$ or $D$ . The discriminant is calculated using the formula $\Delta = b^2 - 4ac$ . If the discriminant is positive, the equation has two distinct solutions. If the discriminant is zero, the equation has one repeated solution. If the discriminant is negative, the equation has no real solutions.

Let's calculate the discriminant for our example equation:

$\Delta = (-20)^2 - 4(1)(100)$ $\Delta = 400 - 400$ $\Delta = 0$

Since the discriminant is zero, we know that the equation has one repeated solution.

Step 3: Solve the Quadratic Equation Using the Quadratic Formula

The quadratic formula is a formula that can be used to solve quadratic equations. It is given by the formula $x = \frac{-b \pm \sqrt{\Delta}}{2a}$ . In our example, we have $a = 1$ , $b = -20$ , and $\Delta = 0$ . Plugging these values into the formula, we get:

$x = \frac{-(-20) \pm \sqrt{0}}{2(1)}$ $x = \frac{20 \pm 0}{2}$ $x = \frac{20}{2}$ $x = 10$

Since the discriminant is zero, we know that the equation has one repeated solution, which is $x = 10$ .

Conclusion

Solving quadratic equations is an important skill that can be used in a variety of fields, including physics, engineering, and economics. In this article, we have guided you through the process of solving a quadratic equation using the discriminant and the quadratic formula. We have used the equation $x^2 - 20x + 100 = 0$ as an example and shown that it has one repeated solution.

Tips and Tricks

Always check the discriminant before using the quadratic formula.
If the discriminant is positive, the equation has two distinct solutions.
If the discriminant is zero, the equation has one repeated solution.
If the discriminant is negative, the equation has no real solutions.

Common Quadratic Equations

$x^2 + 4x + 4 = 0$
$x^2 - 6x + 9 = 0$
$x^2 + 2x + 1 = 0$

Quadratic Formula Derivation

The quadratic formula can be derived by using the method of completing the square. This involves rewriting the quadratic equation in the form $(x + \frac{b}{2a})^2 = \frac{c}{a}$ . The quadratic formula is then obtained by taking the square root of both sides and solving for $x$ .

Real-World Applications

Quadratic equations have many real-world applications, including:

Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
Engineering: Quadratic equations are used to design bridges and other structures.
Economics: Quadratic equations are used to model the behavior of economic systems.

Conclusion

Frequently Asked Questions

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants.

Q: How do I know if a quadratic equation has real solutions?

A: To determine if a quadratic equation has real solutions, you need to calculate the discriminant, which is given by the formula $\Delta = b^2 - 4ac$ . If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one repeated real solution. If the discriminant is negative, the equation has no real solutions.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to solve quadratic equations. It is given by the formula $x = \frac{-b \pm \sqrt{\Delta}}{2a}$ . This formula can be used to find the solutions of a quadratic equation when the discriminant is positive or zero.

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of $a$ , $b$ , and $\Delta$ into the formula. If the discriminant is positive, you will get two distinct solutions. If the discriminant is zero, you will get one repeated solution.

Q: What is the difference between a quadratic equation and a linear equation?

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. A quadratic equation has a highest power of two, while a linear equation has a highest power of one.

Q: Can I use the quadratic formula to solve a quadratic equation with complex solutions?

A: Yes, you can use the quadratic formula to solve a quadratic equation with complex solutions. However, you need to be careful when taking the square root of the discriminant, as it may result in complex numbers.

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, you can use the x-intercepts and the vertex of the parabola to plot the graph. The x-intercepts are the points where the parabola intersects the x-axis, and the vertex is the point where the parabola is at its maximum or minimum value.

Q: What is the vertex of a quadratic equation?

A: The vertex of a quadratic equation is the point where the parabola is at its maximum or minimum value. The vertex can be found using the formula $x = -\frac{b}{2a}$ .

Q: Can I use the quadratic formula to solve a quadratic equation with a fractional coefficient?

A: Yes, you can use the quadratic formula to solve a quadratic equation with a fractional coefficient. However, you need to be careful when simplifying the expression, as it may result in complex fractions.

Q: How do I simplify a quadratic equation?

A: To simplify a quadratic equation, you can use the following steps:

Factor the equation, if possible.
Use the quadratic formula to solve the equation.
Simplify the expression, if necessary.

Q: Can I use the quadratic formula to solve a quadratic equation with a negative coefficient?

A: Yes, you can use the quadratic formula to solve a quadratic equation with a negative coefficient. However, you need to be careful when simplifying the expression, as it may result in complex numbers.

Conclusion

In conclusion, the quadratic equation is a fundamental concept in mathematics that has many real-world applications. In this article, we have answered some frequently asked questions about quadratic equations, including how to determine if a quadratic equation has real solutions, how to use the quadratic formula, and how to graph a quadratic equation. We hope that this article has been helpful in understanding the concept of quadratic equations and how to solve them.