Solve The Equation By Taking Square Roots. Give The Answer In Radical Form, And Then Use A Calculator To Approximate The Solution To Two Decimal Places If Necessary.$5x^2 + 10 = 110$

Feb 26, 2025 by ADMIN 183 views

Introduction

In this article, we will learn how to solve a quadratic equation by taking square roots. We will start with a given equation, $5x^2 + 10 = 110$ , and then use algebraic manipulations to isolate the variable $x$ . We will express the solution in radical form and then use a calculator to approximate the solution to two decimal places if necessary.

Step 1: Subtract 10 from Both Sides

The first step in solving the equation is to isolate the term with the variable $x$ . We can do this by subtracting 10 from both sides of the equation.

5x^2 + 10 - 10 = 110 - 10

This simplifies to:

5x^2 = 100

Step 2: Divide Both Sides by 5

Next, we need to isolate the term with the variable $x$ by dividing both sides of the equation by 5.

\frac{5x^2}{5} = \frac{100}{5}

This simplifies to:

x^2 = 20

Step 3: Take the Square Root of Both Sides

Now that we have isolated the term with the variable $x$ , we can take the square root of both sides of the equation.

\sqrt{x^2} = \sqrt{20}

This simplifies to:

x = \pm \sqrt{20}

Step 4: Simplify the Radical

We can simplify the radical by factoring out a perfect square.

x = \pm \sqrt{4 \cdot 5}

This simplifies to:

x = \pm 2\sqrt{5}

Approximating the Solution

To approximate the solution to two decimal places, we can use a calculator.

x \approx \pm 4.47

Conclusion

In this article, we learned how to solve a quadratic equation by taking square roots. We started with a given equation, $5x^2 + 10 = 110$ , and then used algebraic manipulations to isolate the variable $x$ . We expressed the solution in radical form and then used a calculator to approximate the solution to two decimal places.

Why is it Important to Learn How to Solve Quadratic Equations?

Solving quadratic equations is an important skill in mathematics because it allows us to model real-world problems. Quadratic equations are used to model a wide range of phenomena, from the trajectory of a projectile to the growth of a population. By learning how to solve quadratic equations, we can gain a deeper understanding of the world around us.

Real-World Applications of Quadratic Equations

Quadratic equations have many real-world applications. For example, they are used in physics to model the motion of objects, in engineering to design buildings and bridges, and in economics to model the growth of a company. By learning how to solve quadratic equations, we can gain a deeper understanding of these applications and develop new skills that can be used in a variety of fields.

Common Mistakes to Avoid When Solving Quadratic Equations

When solving quadratic equations, there are several common mistakes to avoid. One of the most common mistakes is to forget to check the solutions for extraneous solutions. Extraneous solutions are solutions that are not valid in the context of the problem. For example, if we are solving the equation $x^2 = 4$ , the solutions are $x = 2$ and $x = -2$ . However, if we are solving the equation $x^2 = -4$ , there are no real solutions. By checking the solutions for extraneous solutions, we can avoid this mistake and ensure that our solutions are valid.

Tips for Solving Quadratic Equations

When solving quadratic equations, there are several tips to keep in mind. One of the most important tips is to always check the solutions for extraneous solutions. Another important tip is to use algebraic manipulations to isolate the variable $x$ . By using algebraic manipulations, we can simplify the equation and make it easier to solve. Finally, by using a calculator to approximate the solution, we can ensure that our solution is accurate to two decimal places.

Conclusion

Introduction

In our previous article, we learned how to solve a quadratic equation by taking square roots. We started with a given equation, $5x^2 + 10 = 110$ , and then used algebraic manipulations to isolate the variable $x$ . We expressed the solution in radical form and then used a calculator to approximate the solution to two decimal places. In this article, we will answer some common questions that students may have when solving quadratic equations.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means that 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, we need to check the discriminant, which is the expression under the square root in the quadratic formula. If the discriminant is positive, then the equation has two real solutions. If the discriminant is zero, then the equation has one real solution. If the discriminant is negative, then 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:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: How do I use the quadratic formula to solve a quadratic equation?

A: To use the quadratic formula to solve a quadratic equation, we need to plug in the values of $a$ , $b$ , and $c$ into the formula. We then simplify the expression under the square root and solve for $x$ .

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. In other words, a quadratic equation has a squared variable, while a linear equation does not.

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

A: Yes, you can use the quadratic formula to solve a quadratic equation with a negative leading coefficient. However, you will need to take the square root of a negative number, which will result in complex solutions.

Q: What is the difference between a quadratic equation and a polynomial equation of degree three?

A: A quadratic equation is a polynomial equation of degree two, while a polynomial equation of degree three is a cubic equation. A cubic equation has a cubed variable, while a quadratic equation has a squared variable.

Q: Can I use the quadratic formula to solve a quadratic equation with a variable in the denominator?

A: No, you cannot use the quadratic formula to solve a quadratic equation with a variable in the denominator. This is because the quadratic formula assumes that the denominator is a constant.

Conclusion

In conclusion, solving quadratic equations is an important skill in mathematics that has many real-world applications. By learning how to solve quadratic equations, we can gain a deeper understanding of the world around us and develop new skills that can be used in a variety of fields. By following the tips and avoiding the common mistakes outlined in this article, we can ensure that our solutions are accurate and valid.