Setting the Equation Equal to 7

Oct 13, 2025 by miodeya.com 27 views

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will explore the solutions to a given quadratic equation and determine which of the provided options are correct.

The Given Equation

The given equation is $${(5x+1)^2 = 7}$$. To solve this equation, we need to expand the left-hand side and then isolate the variable $x$ .

Expanding the Left-Hand Side

To expand the left-hand side of the equation, we need to use the formula $(a+b)^2 = a^2 + 2ab + b^2$ . In this case, $a = 5x$ and $b = 1$ . Therefore, we have:

{(5x+1)^2 = (5x)^2 + 2(5x)(1) + 1^2}

Simplifying the Equation

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

{(5x)^2 + 2(5x)(1) + 1^2 = 25x^2 + 10x + 1}

Setting the Equation Equal to 7

Since the left-hand side of the equation is equal to 7, we can set the simplified equation equal to 7:

{25x^2 + 10x + 1 = 7}

Subtracting 7 from Both Sides

To isolate the quadratic term, we need to subtract 7 from both sides of the equation:

{25x^2 + 10x - 6 = 0}

Solving the Quadratic Equation

Now, we can solve the quadratic equation using the quadratic formula:

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

In this case, $a = 25$ , $b = 10$ , and $c = -6$ . Plugging these values into the quadratic formula, we get:

{x = \frac{-10 \pm \sqrt{10^2 - 4(25)(-6)}}{2(25)}}

Simplifying the Quadratic Formula

Now, we can simplify the quadratic formula by evaluating the expression inside the square root:

{x = \frac{-10 \pm \sqrt{100 + 600}}{50}}

{x = \frac{-10 \pm \sqrt{700}}{50}}

Simplifying the Square Root

The square root of 700 can be simplified as follows:

{\sqrt{700} = \sqrt{100 \cdot 7} = 10\sqrt{7}}

Simplifying the Quadratic Formula

Now, we can simplify the quadratic formula by plugging in the simplified square root:

{x = \frac{-10 \pm 10\sqrt{7}}{50}}

Simplifying the Fraction

Now, we can simplify the fraction by dividing both the numerator and the denominator by 10:

{x = \frac{-1 \pm \sqrt{7}}{5}}

Evaluating the Two Possible Solutions

Now, we have two possible solutions for the quadratic equation:

{x = \frac{-1 + \sqrt{7}}{5}}

{x = \frac{-1 - \sqrt{7}}{5}}

Comparing the Solutions to the Given Options

Now, we can compare the two possible solutions to the given options:

A. $${x = -\frac{\sqrt{6}}{5}}$$

B. $${x = \frac{\sqrt{7}}{5} - 1}$$

C. $${x = \frac{-\sqrt{7}}{5} - 1}$$

D. $${x = \frac{-\sqrt{7}}{5} + 1}$$

Conclusion

Based on our calculations, we can see that options B and C are not correct. Option A is also not correct, as the square root of 6 is not equal to the square root of 7. However, option D is correct, as it matches one of the two possible solutions we obtained.

The final answer is: $\boxed{D}$

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will provide a Q&A guide to help students understand quadratic equations and how to solve them.

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. It is typically written in the form $ax^2 + bx + c = 0$ , 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 completing the square. The method you choose will depend on the specific equation and the values of the coefficients.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to solve a quadratic equation. It is given by:

{x = \frac{-b \pm \sqrt{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. You will then get two possible solutions for the equation.

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 "plus" solution and the "minus" solution. They are given by:

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

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

Q: How do I determine which solution is correct?

A: To determine which solution is correct, you need to plug in the values of $x$ into the original equation and see if it is true. If it is true, then that solution is correct.

Q: What is the significance of the discriminant in the quadratic formula?

A: The discriminant is the expression under the square root in the quadratic formula, which is given by $b^2 - 4ac$ . If the discriminant is positive, then the equation has two distinct 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: How do I determine the number of solutions to a quadratic equation?

A: To determine the number of solutions to a quadratic equation, you need to look at the discriminant. If the discriminant is positive, then the equation has two distinct 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 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: 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 table of values to plot the points on the graph.

Conclusion

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. By understanding the quadratic formula and how to use it, you can solve quadratic equations and determine the number of solutions to an equation.