Using a Graphing Calculator to Solve an Equation

Oct 13, 2025 by magicalceylontours.com 92 views

Introduction

Equations involving radicals and exponents can be challenging to solve, but with the right techniques and tools, we can find the solutions. In this article, we will solve four equations involving radicals and exponents, and then use a graphing calculator to solve a fifth equation.

Equation 1: Solving a Linear Equation with a Radical

The first equation we will solve is:

$\sqrt{x+13} - 7 = 3$

To solve this equation, we need to isolate the radical expression. We can do this by adding 7 to both sides of the equation:

$\sqrt{x+13} = 10$

Next, we can square both sides of the equation to eliminate the radical:

$x+13 = 100$

Subtracting 13 from both sides gives us:

$x = 87$

Therefore, the solution to the first equation is $x = 87$ .

Equation 2: Solving an Exponential Equation

The second equation we will solve is:

$(2x+1)^{\frac{2}{3}} = 25$

To solve this equation, we need to isolate the exponential expression. We can do this by cubing both sides of the equation:

$(2x+1) = 25^{\frac{3}{2}}$

Simplifying the right-hand side gives us:

$(2x+1) = 125$

Subtracting 1 from both sides gives us:

$2x = 124$

Dividing both sides by 2 gives us:

$x = 62$

Therefore, the solution to the second equation is $x = 62$ .

Equation 3: Solving a Radical Equation with Two Radicals

The third equation we will solve is:

$\sqrt{x+7} - \sqrt{x} = 1$

To solve this equation, we need to isolate one of the radical expressions. We can do this by adding $\sqrt{x}$ to both sides of the equation:

$\sqrt{x+7} = \sqrt{x} + 1$

Next, we can square both sides of the equation to eliminate the radicals:

$x+7 = x + 2\sqrt{x} + 1$

Subtracting $x$ and 7 from both sides gives us:

$0 = 2\sqrt{x} - 6$

Dividing both sides by 2 gives us:

$0 = \sqrt{x} - 3$

Adding 3 to both sides gives us:

$3 = \sqrt{x}$

Squaring both sides gives us:

$9 = x$

Therefore, the solution to the third equation is $x = 9$ .

Equation 4: Solving a Radical Equation with a Fractional Index

The fourth equation we will solve is:

$\sqrt[4]{7x+2} + 1 = 3$

To solve this equation, we need to isolate the radical expression. We can do this by subtracting 1 from both sides of the equation:

$\sqrt[4]{7x+2} = 2$

Next, we can raise both sides of the equation to the fourth power to eliminate the radical:

$7x+2 = 16$

Subtracting 2 from both sides gives us:

$7x = 14$

Dividing both sides by 7 gives us:

$x = 2$

Therefore, the solution to the fourth equation is $x = 2$ .

Using a Graphing Calculator to Solve an Equation

The fifth equation we will solve is:

$\sqrt{5x+1} = \sqrt{6x-6}$

To solve this equation, we can use a graphing calculator to graph the two functions on either side of the equation. We can then find the intersection point of the two graphs, which will give us the solution to the equation.

Using a graphing calculator, we can graph the two functions:

$y = \sqrt{5x+1}$
$y = \sqrt{6x-6}$

The graph shows that the two functions intersect at $x = 1$ .

Therefore, the solution to the fifth equation is $x = 1$ .

Conclusion

Introduction

In our previous article, we solved four equations involving radicals and exponents, and then used a graphing calculator to solve a fifth equation. In this article, we will answer some common questions that students often have when solving equations with radicals and exponents.

Q: What is the difference between a radical and an exponent?

A: A radical is a symbol that represents the square root of a number, while an exponent is a number that represents the power to which a base number is raised. For example, $\sqrt{x}$ is a radical, while $x^2$ is an exponent.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to find the largest perfect square that divides the number inside the radical. You can then take the square root of that perfect square and multiply it by the remaining number inside the radical. For example, $\sqrt{12}$ can be simplified as $\sqrt{4 \times 3}$ , which is equal to $2\sqrt{3}$ .

Q: How do I solve an equation with a radical on both sides?

A: To solve an equation with a radical on both sides, you need to isolate one of the radical expressions. You can do this by adding or subtracting the same value to both sides of the equation. For example, if you have the equation $\sqrt{x} + 3 = 5$ , you can subtract 3 from both sides to get $\sqrt{x} = 2$ .

Q: How do I solve an equation with an exponent on both sides?

A: To solve an equation with an exponent on both sides, you need to isolate one of the exponential expressions. You can do this by taking the logarithm of both sides of the equation. For example, if you have the equation $x^2 = 16$ , you can take the logarithm of both sides to get $\log(x^2) = \log(16)$ .

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

A: A linear equation is an equation in which the highest power of the variable is 1, while a quadratic equation is an equation in which the highest power of the variable is 2. For example, $x + 2 = 5$ is a linear equation, while $x^2 + 2x + 1 = 0$ is a quadratic equation.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula, which is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . You can also use factoring, which involves finding two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term.

Q: What is the difference between a rational equation and an irrational equation?

A: A rational equation is an equation in which the variable is in the numerator or denominator of a fraction, while an irrational equation is an equation in which the variable is not in the numerator or denominator of a fraction. For example, $\frac{x}{2} = 3$ is a rational equation, while $\sqrt{x} = 2$ is an irrational equation.

Q: How do I solve a rational equation?

A: To solve a rational equation, you need to eliminate the fractions by multiplying both sides of the equation by the least common multiple of the denominators. You can then solve the resulting equation using the techniques we discussed earlier.

Conclusion

In this article, we answered some common questions that students often have when solving equations with radicals and exponents. We discussed the difference between radicals and exponents, how to simplify radical expressions, how to solve equations with radicals and exponents on both sides, and how to solve quadratic equations. We also discussed the difference between rational and irrational equations and how to solve rational equations.