Solve The Equation $5x^2 - 2x - 5 = 0$ To The Nearest Tenth.$x =$ $\square$

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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 focus on solving the quadratic equation $5x^2 - 2x - 5 = 0$ to the nearest tenth.

What is a Quadratic Equation?

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

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

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

This formula will be used to solve the given quadratic equation.

Solving the Quadratic Equation

Now, let's apply the quadratic formula to solve the equation $5x^2 - 2x - 5 = 0$.

Step 1: Identify the coefficients

In the given equation, the coefficients are a = 5, b = -2, and c = -5.

Step 2: Plug the coefficients into the quadratic formula

Substituting the values of a, b, and c into the quadratic formula, we get:

x=−(−2)±(−2)2−4(5)(−5)2(5)x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(5)(-5)}}{2(5)}

Step 3: Simplify the expression

Simplifying the expression inside the square root, we get:

x=2±4+10010x = \frac{2 \pm \sqrt{4 + 100}}{10}

x=2±10410x = \frac{2 \pm \sqrt{104}}{10}

Step 4: Simplify the square root

The square root of 104 can be simplified as:

104=4×26=226\sqrt{104} = \sqrt{4 \times 26} = 2\sqrt{26}

Substituting this value back into the expression, we get:

x=2±22610x = \frac{2 \pm 2\sqrt{26}}{10}

Step 5: Simplify the expression

Simplifying the expression further, we get:

x=1±265x = \frac{1 \pm \sqrt{26}}{5}

Solving for x

Now, we have two possible solutions for x:

x=1+265x = \frac{1 + \sqrt{26}}{5}

x=1−265x = \frac{1 - \sqrt{26}}{5}

Rounding to the Nearest Tenth

To solve the equation to the nearest tenth, we need to round the values of x to one decimal place.

Solution 1

Rounding the value of x to the nearest tenth, we get:

x≈1+5.15x \approx \frac{1 + 5.1}{5}

x≈6.15x \approx \frac{6.1}{5}

x≈1.2x \approx 1.2

Solution 2

Rounding the value of x to the nearest tenth, we get:

x≈1−5.15x \approx \frac{1 - 5.1}{5}

x≈−4.15x \approx \frac{-4.1}{5}

x≈−0.8x \approx -0.8

Conclusion

In this article, we solved the quadratic equation $5x^2 - 2x - 5 = 0$ to the nearest tenth using the quadratic formula. We obtained two possible solutions for x, which were rounded to one decimal place. The solutions are x ≈ 1.2 and x ≈ -0.8.

Final Answer

The final answer is: 1.2\boxed{1.2}

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Introduction

In our previous article, we solved the quadratic equation $5x^2 - 2x - 5 = 0$ to the nearest tenth using the quadratic formula. In this article, we will address some frequently asked questions related to quadratic equations.

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 (in this case, x) is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where a, b, and c are constants.

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

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

Q: How do I apply the quadratic formula?

A: To apply the quadratic formula, you need to identify the coefficients a, b, and c in the quadratic equation. Then, plug these values into the quadratic formula and simplify the expression.

Q: What is the difference between the two solutions obtained from the quadratic formula?

A: The two solutions obtained from the quadratic formula are called the distinct roots of the quadratic equation. These roots are obtained by using the plus and minus signs in the quadratic formula.

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

A: No, a quadratic equation can have at most two solutions. This is because the quadratic formula produces two distinct roots, and there are no other possible solutions.

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

A: To determine the number of solutions of a quadratic equation, you need to examine the discriminant, which is the expression under the square root in 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.

Q: What is the discriminant?

A: The discriminant is the expression under the square root in the quadratic formula, which is given by:

b2−4acb^2 - 4ac

Q: How do I simplify the discriminant?

A: To simplify the discriminant, you need to expand the expression and combine like terms.

Q: Can a quadratic equation have complex solutions?

A: Yes, a quadratic equation can have complex solutions. This occurs when the discriminant is negative, and the quadratic formula produces complex roots.

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

A: To find the complex solutions of a quadratic equation, you need to use the quadratic formula and simplify the expression. The complex solutions will be in the form of a + bi, where a and b are real numbers and i is the imaginary unit.

Q: What is the imaginary unit?

A: The imaginary unit is a mathematical concept that is used to represent complex numbers. It is denoted by i and is defined as the square root of -1.

Conclusion

In this article, we addressed some frequently asked questions related to quadratic equations. We covered topics such as the quadratic formula, distinct roots, discriminant, and complex solutions. We hope that this article has provided you with a better understanding of quadratic equations and how to solve them.

Final Answer

The final answer is: 1.2\boxed{1.2}