Solve The Following Equations:1. \[$2 \sqrt{x-3} + 3 = 0\$\]2. \[$\frac{\sqrt{x}}{\sqrt{2}} = 3 \sqrt{2}\$\]3. \[$6 \sqrt{x^2-9} = 4\$\]4. \[$8 \sqrt{x+5} - X = 3\$\]5. \[$19x + \sqrt{5-x} + 1 = 0\$\]6. \[$x

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Introduction

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Equations involving square roots can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will explore six different equations involving square roots and provide step-by-step solutions to each of them.

Equation 1: $2 \sqrt{x-3} + 3 = 0$

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Step 1: Isolate the Square Root Term

To solve the equation, we need to isolate the square root term. We can do this by subtracting 3 from both sides of the equation.

$$2 \sqrt{x-3} = -3$$

Step 2: Divide Both Sides by 2

Next, we divide both sides of the equation by 2 to get:

$$\sqrt{x-3} = -\frac{3}{2}$$

Step 3: Square Both Sides

Now, we square both sides of the equation to get rid of the square root.

$$(\sqrt{x-3})^2 = \left(-\frac{3}{2}\right)^2$$

This simplifies to:

$$x-3 = \frac{9}{4}$$

Step 4: Add 3 to Both Sides

Finally, we add 3 to both sides of the equation to solve for x.

$$x = \frac{9}{4} + 3$$

$$x = \frac{9+12}{4}$$

$$x = \frac{21}{4}$$

The final answer is $\boxed{\frac{21}{4}}$.

Equation 2: $\frac{\sqrt{x}}{\sqrt{2}} = 3 \sqrt{2}$

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Step 1: Multiply Both Sides by $\sqrt{2}$

To solve the equation, we multiply both sides by $\sqrt{2}$ to get rid of the fraction.

$$\sqrt{x} = 3 \sqrt{2} \cdot \sqrt{2}$$

This simplifies to:

$$\sqrt{x} = 6$$

Step 2: Square Both Sides

Now, we square both sides of the equation to get rid of the square root.

$$(\sqrt{x})^2 = 6^2$$

This simplifies to:

$$x = 36$$

The final answer is $\boxed{36}$.

Equation 3: $6 \sqrt{x^2-9} = 4$

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Step 1: Divide Both Sides by 6

To solve the equation, we divide both sides by 6 to get:

$$\sqrt{x^2-9} = \frac{4}{6}$$

This simplifies to:

$$\sqrt{x^2-9} = \frac{2}{3}$$

Step 2: Square Both Sides

Now, we square both sides of the equation to get rid of the square root.

$$(\sqrt{x^2-9})^2 = \left(\frac{2}{3}\right)^2$$

This simplifies to:

$$x^2-9 = \frac{4}{9}$$

Step 3: Add 9 to Both Sides

Next, we add 9 to both sides of the equation to get:

$$x^2 = \frac{4}{9} + 9$$

This simplifies to:

$$x^2 = \frac{4+81}{9}$$

$$x^2 = \frac{85}{9}$$

Step 4: Take the Square Root of Both Sides

Finally, we take the square root of both sides of the equation to solve for x.

$$x = \pm \sqrt{\frac{85}{9}}$$

The final answer is $\boxed{\pm \sqrt{\frac{85}{9}}}$.

Equation 4: $8 \sqrt{x+5} - x = 3$

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Step 1: Add x to Both Sides

To solve the equation, we add x to both sides to get:

$$8 \sqrt{x+5} = x + 3$$

Step 2: Divide Both Sides by 8

Next, we divide both sides by 8 to get:

$$\sqrt{x+5} = \frac{x+3}{8}$$

Step 3: Square Both Sides

Now, we square both sides of the equation to get rid of the square root.

$$(\sqrt{x+5})^2 = \left(\frac{x+3}{8}\right)^2$$

This simplifies to:

$$x+5 = \frac{(x+3)^2}{64}$$

Step 4: Multiply Both Sides by 64

Next, we multiply both sides by 64 to get:

$$64(x+5) = (x+3)^2$$

This simplifies to:

$$64x + 320 = x^2 + 6x + 9$$

Step 5: Rearrange the Equation

Now, we rearrange the equation to get a quadratic equation in standard form.

$$x^2 - 58x - 311 = 0$$

Step 6: Solve the Quadratic Equation

We can solve the quadratic equation using the quadratic formula.

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

In this case, a = 1, b = -58, and c = -311.

$$x = \frac{58 \pm \sqrt{(-58)^2 - 4(1)(-311)}}{2(1)}$$

This simplifies to:

$$x = \frac{58 \pm \sqrt{3364 + 1244}}{2}$$

$$x = \frac{58 \pm \sqrt{4608}}{2}$$

$$x = \frac{58 \pm 68}{2}$$

This gives us two possible solutions:

$$x = \frac{58 + 68}{2}$$

$$x = \frac{126}{2}$$

$$x = 63$$

$$x = \frac{58 - 68}{2}$$

$$x = \frac{-10}{2}$$

$$x = -5$$

The final answer is $\boxed{63}$ or $\boxed{-5}$.

Equation 5: $19x + \sqrt{5-x} + 1 = 0$

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Step 1: Subtract 1 from Both Sides

To solve the equation, we subtract 1 from both sides to get:

$$19x + \sqrt{5-x} = -1$$

Step 2: Subtract 19x from Both Sides

Next, we subtract 19x from both sides to get:

$$\sqrt{5-x} = -19x - 1$$

Step 3: Square Both Sides

Now, we square both sides of the equation to get rid of the square root.

$$(\sqrt{5-x})^2 = (-19x - 1)^2$$

This simplifies to:

$$5-x = 361x^2 + 38x + 1$$

Step 4: Rearrange the Equation

Next, we rearrange the equation to get a quadratic equation in standard form.

$$361x^2 + 43x + 6 = 0$$

Step 5: Solve the Quadratic Equation

We can solve the quadratic equation using the quadratic formula.

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

In this case, a = 361, b = 43, and c = 6.

$$x = \frac{-43 \pm \sqrt{(43)^2 - 4(361)(6)}}{2(361)}$$

This simplifies to:

$$x = \frac{-43 \pm \sqrt{1849 - 8664}}{722}$$

$$x = \frac{-43 \pm \sqrt{-6815}}{722}$$

Since the discriminant is negative, there are no real solutions to the equation.

The final answer is $\boxed{No real solutions}$.

Equation 6: $x \sqrt{x+1} = 4$

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Step 1: Divide Both Sides by $\sqrt{x+1}$

To solve the equation, we divide both sides by $\sqrt{x+1}$ to get:

$$x = \frac{4}{\sqrt{x+1}}$$

Step 2: Square Both Sides

Next, we square both sides of the equation to get rid of the fraction.

$$(x)^2 = \left(\frac{4}{\sqrt{x+1}}\right)^2$$

This simplifies to:

$$x^2 = \frac{16}{x+1}$$

Step 3: Cross Multiply

Now, we cross multiply to get rid of the fraction.

$$x^2(x+1) = 16$$

This simplifies to:

$$x^3 + x^2 = 16$$

Step 4: Rearrange the Equation

Next, we rearrange the equation to get a cubic equation in standard form.

$$x^3 + x^2 - 16 = 0$$

Step 5: Solve the Cubic Equation

We can solve the cubic equation using various methods, such as the rational root theorem or synthetic division.

One possible solution is x = 2.

To verify

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Introduction

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Solving equations involving square roots can be a challenging task, but with the right approach, it can be done with ease. In this article, we will provide a Q&A guide to help you understand the concepts and techniques involved in solving equations involving square roots.

Q: What is the first step in solving an equation involving a square root?

A: The first step in solving an equation involving a square root is to isolate the square root term. This can be done by subtracting or adding a constant to both sides of the equation.

Q: How do I get rid of the square root in an equation?

A: To get rid of the square root in an equation, you can square both sides of the equation. This will eliminate the square root and allow you to solve for the variable.

Q: What is the difference between a square root and a cube root?

A: A square root is a root that is raised to the power of 1/2, while a cube root is a root that is raised to the power of 1/3. In other words, a square root is the inverse operation of squaring a number, while a cube root is the inverse operation of cubing a number.

Q: How do I solve an equation involving a square root and a fraction?

A: To solve an equation involving a square root and a fraction, you can multiply both sides of the equation by the denominator of the fraction. This will eliminate the fraction and allow you to solve for the variable.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that is 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, you need to identify the values of a, b, and c in the equation. Then, you can plug these values into the quadratic formula and simplify to find the solutions.

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

A: A rational root is a root that can be expressed as a fraction, while an irrational root is a root that cannot be expressed as a fraction. In other words, a rational root is a root that can be written in the form p/q, where p and q are integers, while an irrational root is a root that cannot be written in this form.

Q: How do I determine if a root is rational or irrational?

A: To determine if a root is rational or irrational, you can use the rational root theorem. This theorem states that if a polynomial equation has a rational root, then that root must be a factor of the constant term.

Q: What is the rational root theorem?

A: The rational root theorem is a theorem that states that if a polynomial equation has a rational root, then that root must be a factor of the constant term. In other words, if a polynomial equation has a rational root, then that root must be a number that divides the constant term.

Q: How do I use the rational root theorem to find the roots of a polynomial equation?

A: To use the rational root theorem to find the roots of a polynomial equation, you need to identify the factors of the constant term. Then, you can use these factors to find the possible rational roots of the equation.

Q: What is the difference between a real root and an imaginary root?

A: A real root is a root that is a real number, while an imaginary root is a root that is a complex number. In other words, a real root is a root that can be expressed as a number on the number line, while an imaginary root is a root that cannot be expressed in this way.

Q: How do I determine if a root is real or imaginary?

A: To determine if a root is real or imaginary, you can use the discriminant. The discriminant is a value that is calculated from the coefficients of the polynomial equation. If the discriminant is positive, then the root is real. If the discriminant is negative, then the root is imaginary.

Q: What is the discriminant?

A: The discriminant is a value that is calculated from the coefficients of a polynomial equation. It is given by:

$$D = b^2 - 4ac$$

Q: How do I use the discriminant to determine if a root is real or imaginary?

A: To use the discriminant to determine if a root is real or imaginary, you need to calculate the discriminant and then check if it is positive or negative. If the discriminant is positive, then the root is real. If the discriminant is negative, then the root is imaginary.

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

A: A linear equation is an equation that has a degree of 1, while a quadratic equation is an equation that has a degree of 2. In other words, a linear equation is an equation that can be written in the form ax + b = 0, while a quadratic equation is an equation that can be written in the form ax^2 + bx + c = 0.

Q: How do I solve a linear equation?

A: To solve a linear equation, you can use the following steps:

1. Add or subtract a constant to both sides of the equation to isolate the variable.
2. Divide both sides of the equation by the coefficient of the variable to solve for the variable.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the following steps:

1. Factor the quadratic expression, if possible.
2. Use the quadratic formula to find the solutions.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that is 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, you need to identify the values of a, b, and c in the equation. Then, you can plug these values into the quadratic formula and simplify to find the solutions.

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

A: A rational root is a root that can be expressed as a fraction, while an irrational root is a root that cannot be expressed as a fraction. In other words, a rational root is a root that can be written in the form p/q, where p and q are integers, while an irrational root is a root that cannot be written in this form.

Q: How do I determine if a root is rational or irrational?

A: To determine if a root is rational or irrational, you can use the rational root theorem. This theorem states that if a polynomial equation has a rational root, then that root must be a factor of the constant term.

Q: What is the rational root theorem?

A: The rational root theorem is a theorem that states that if a polynomial equation has a rational root, then that root must be a factor of the constant term. In other words, if a polynomial equation has a rational root, then that root must be a number that divides the constant term.

Q: How do I use the rational root theorem to find the roots of a polynomial equation?

A: To use the rational root theorem to find the roots of a polynomial equation, you need to identify the factors of the constant term. Then, you can use these factors to find the possible rational roots of the equation.

Q: What is the difference between a real root and an imaginary root?

A: A real root is a root that is a real number, while an imaginary root is a root that is a complex number. In other words, a real root is a root that can be expressed as a number on the number line, while an imaginary root is a root that cannot be expressed in this way.

Q: How do I determine if a root is real or imaginary?

A: To determine if a root is real or imaginary, you can use the discriminant. The discriminant is a value that is calculated from the coefficients of the polynomial equation. If the discriminant is positive, then the root is real. If the discriminant is negative, then the root is imaginary.

Q: What is the discriminant?

A: The discriminant is a value that is calculated from the coefficients of a polynomial equation. It is given by:

$$D = b^2 - 4ac$$

Q: How do I use the discriminant to determine if a root is real or imaginary?

A: To use the discriminant to determine if a root is real or imaginary, you need to calculate the discriminant and then check if it is positive or negative. If the discriminant is positive, then the root is real. If the discriminant is negative, then the root is imaginary.

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

A: A linear equation is an equation that has a degree of 1, while a quadratic equation is an equation that has a degree of 2. In other words, a linear equation is an equation that can be written in the form ax + b = 0, while a quadratic equation is an equation that can be written in the form ax^2 + bx + c = 0.

Q: How do I solve a linear equation?

A: To solve a linear equation, you can use the following steps:

1. Add or subtract a constant to both sides of the equation to isolate the variable.