Solve The Equation 3.5 X 2 + 7.5 X − 2.2 = − 2.9 3.5x^2 + 7.5x - 2.2 = -2.9 3.5 X 2 + 7.5 X − 2.2 = − 2.9 Using A Calculator. Input Your Solutions Below In Any Order And Round To Three Decimal Places If Necessary. X 1 = X_1 = X 1 ​ = □ \square □ Or X 2 = X_2 = X 2 ​ = □ \square □

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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 explore how to solve a quadratic equation using a calculator. We will use the given equation 3.5x2+7.5x−2.2=−2.93.5x^2 + 7.5x - 2.2 = -2.9 as an example and provide step-by-step instructions on how to input the equation into a calculator and find the solutions.

Understanding the Equation

Before we begin, let's take a closer look at the given equation:

3.5x2+7.5x−2.2=−2.93.5x^2 + 7.5x - 2.2 = -2.9

This is a quadratic equation in the form of ax2+bx+c=dax^2 + bx + c = d, where a=3.5a = 3.5, b=7.5b = 7.5, c=−2.2c = -2.2, and d=−2.9d = -2.9. Our goal is to find the values of xx that satisfy this equation.

Using a Calculator to Solve the Equation

To solve the equation using a calculator, we need to follow these steps:

Step 1: Enter the Equation into the Calculator

First, we need to enter the equation into the calculator. We can do this by typing in the values of aa, bb, cc, and dd into the calculator's equation solver.

3.5x^2 + 7.5x - 2.2 = -2.9

Step 2: Set the Equation to Zero

Next, we need to set the equation to zero by subtracting dd from both sides of the equation. This will give us a quadratic equation in the form of ax2+bx+c=0ax^2 + bx + c = 0.

3.5x^2 + 7.5x - 2.2 + 2.9 = 0

Step 3: Simplify the Equation

Now, we can simplify the equation by combining like terms.

3.5x^2 + 7.5x + 0.7 = 0

Step 4: Use the Calculator's Quadratic Formula

Most calculators have a built-in quadratic formula that we can use to solve the equation. The quadratic formula is:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

We can enter this formula into the calculator and plug in the values of aa, bb, and cc.

Step 5: Solve for xx

Finally, we can solve for xx by using the quadratic formula.

x = \frac{-7.5 \pm \sqrt{7.5^2 - 4(3.5)(0.7)}}{2(3.5)}

Calculating the Solutions

Now that we have the quadratic formula, we can calculate the solutions to the equation.

x = \frac{-7.5 \pm \sqrt{56.25 - 9.8}}{7}

x = \frac{-7.5 \pm \sqrt{46.45}}{7}

x = \frac{-7.5 \pm 6.8}{7}

Finding the Solutions

Now that we have the solutions, we can find the values of xx that satisfy the equation.

x_1 = \frac{-7.5 + 6.8}{7}

x_1 = \frac{-0.7}{7}

x_1 = -0.1

x_2 = \frac{-7.5 - 6.8}{7}

x_2 = \frac{-14.3}{7}

x_2 = -2.0

Conclusion

In this article, we have shown how to solve a quadratic equation using a calculator. We used the given equation 3.5x2+7.5x−2.2=−2.93.5x^2 + 7.5x - 2.2 = -2.9 as an example and provided step-by-step instructions on how to input the equation into a calculator and find the solutions. We also calculated the solutions to the equation and found the values of xx that satisfy the equation.

Final Answer

The final answer is:

x1=−0.1x_1 = -0.1 x2=−2.0x_2 = -2.0

Note: The solutions are rounded to three decimal places if necessary.

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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 (usually x) is two. It is in the form of ax^2 + bx + c = 0, where a, b, and c are constants.

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including factoring, using the quadratic formula, and graphing. The quadratic formula is the most common method and is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that is used to solve quadratic equations. It is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

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 c into the formula. Then, simplify the expression and solve for x.

Q: What are the solutions to a quadratic equation?

A: The solutions to a quadratic equation are the values of x that satisfy the equation. They can be real or complex numbers.

Q: How do I determine the number of solutions to a quadratic equation?

A: The number of solutions to a quadratic equation can be determined by the discriminant (b^2 - 4ac). If the discriminant is positive, there are two real solutions. If the discriminant is zero, there is one real solution. If the discriminant is negative, there are two complex solutions.

Q: What is the discriminant?

A: The discriminant is a value that is used to determine the number of solutions to a quadratic equation. It is given by b^2 - 4ac.

Q: How do I calculate the discriminant?

A: To calculate the discriminant, you need to plug in the values of a, b, and c into the formula b^2 - 4ac.

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 parabolic shape, while a linear equation has a straight line shape.

Q: Can a quadratic equation have more than two solutions?

A: No, a quadratic equation can have at most two solutions.

Q: Can a quadratic equation have no solutions?

A: Yes, a quadratic equation can have no solutions if the discriminant is negative.

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, you need to use a graphing calculator or a computer program. You can also use a graphing app on your phone or tablet.

Q: What is the vertex of a quadratic equation?

A: The vertex of a quadratic equation is the point on the graph where the parabola changes direction. It is given by the formula x = -b / 2a.

Q: How do I find the vertex of a quadratic equation?

A: To find the vertex of a quadratic equation, you need to plug in the values of a, b, and c into the formula x = -b / 2a.

Q: What is the axis of symmetry of a quadratic equation?

A: The axis of symmetry of a quadratic equation is a vertical line that passes through the vertex of the parabola. It is given by the formula x = -b / 2a.

Q: How do I find the axis of symmetry of a quadratic equation?

A: To find the axis of symmetry of a quadratic equation, you need to plug in the values of a, b, and c into the formula x = -b / 2a.

Conclusion

In this article, we have answered some of the most frequently asked questions about quadratic equations. We have covered topics such as the quadratic formula, the discriminant, and graphing quadratic equations. We hope that this article has been helpful in understanding quadratic equations and their applications.

Final Answer

The final answer is:

  • The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a.
  • The discriminant is b^2 - 4ac.
  • The vertex of a quadratic equation is given by x = -b / 2a.
  • The axis of symmetry of a quadratic equation is given by x = -b / 2a.