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To solve an equation that contains a variable under a square root, follow a four-step procedure:

**Step 1.** **Isolate the radical** on one side of the equation. If the square root expression is not already by itself, use addition, subtraction, or other algebraic operations to move all other terms to the opposite side.

**Step 2.** **Square both sides** of the equation. Applying the Squaring Property of Square Roots removes the radical from the isolated side, converting the equation into a standard polynomial equation.

**Step 3.** **Solve the new equation.** Use appropriate algebraic techniques (such as combining like terms, factoring, or applying properties of equality) to find the value(s) of the variable.

**Step 4.** **Check the answer.** Substitute each candidate solution back into the **original** radical equation and verify that it produces a true statement. Any value that does not satisfy the original equation is an extraneous solution and must be discarded.

How to Solve a Radical Equation

Solve the radical equation $$\sqrt{5n - 4} - 9 = 0$$ by applying the four-step procedure for radical equations.

**Step 1 — Isolate the radical.** The square root is not alone on the left side because of the $$-9$$ term. Add $$9$$ to both sides to move it:

$$\sqrt{5n - 4} - 9 + 9 = 0 + 9$$

$$\sqrt{5n - 4} = 9$$

**Step 2 — Square both sides.** Apply the Squaring Property to remove the radical:

$$(\sqrt{5n - 4})^2 = 9^2$$

$$5n - 4 = 81$$

**Step 3 — Solve the new equation.** Add $$4$$ to both sides: $$5n = 85$$. Divide both sides by $$5$$: $$n = 17$$.

**Step 4 — Check.** Substitute $$n = 17$$ into the original equation:

$$\sqrt{5(17) - 4} - 9 = \sqrt{85 - 4} - 9 = \sqrt{81} - 9 = 9 - 9 = 0$$

Since $$0 = 0$$ is true, $$n = 17$$ is confirmed as the solution. Unlike the simpler example $$\sqrt{2x - 1} = 7$$ where the radical is already isolated, this equation requires an algebraic step (adding $$9$$ to both sides) before the squaring step can be applied.

Solving $$\sqrt{5n-4} - 9 = 0$$

Solve the radical equation $$\sqrt{3y + 5} + 2 = 5$$ by applying the four-step procedure.

**Step 1 — Isolate the radical.** Subtract $$2$$ from both sides to get the square root by itself:

$$\sqrt{3y + 5} + 2 - 2 = 5 - 2$$

$$\sqrt{3y + 5} = 3$$

**Step 2 — Square both sides.** Apply the Squaring Property:

$$(\sqrt{3y + 5})^2 = 3^2$$

$$3y + 5 = 9$$

**Step 3 — Solve the new equation.** Subtract $$5$$ from both sides: $$3y = 4$$. Divide both sides by $$3$$: $$y = \frac{4}{3}$$.

**Step 4 — Check.** Substitute $$y = \frac{4}{3}$$ into the original equation:

$$\sqrt{3 \cdot \frac{4}{3} + 5} + 2 = \sqrt{4 + 5} + 2 = \sqrt{9} + 2 = 3 + 2 = 5$$

Since $$5 = 5$$ is true, $$y = \frac{4}{3}$$ is confirmed as the solution. This example shows that radical equations can produce fractional solutions. In the check step, substituting the fraction $$\frac{4}{3}$$ into $$3 \cdot \frac{4}{3}$$ conveniently simplifies to the integer $$4$$, making the verification straightforward.

Solving $$\sqrt{3y+5} + 2 = 5$$

A radical equation has **no solution** when isolating the square root on one side of the equation results in the radical equaling a negative number. Because the radical sign $$\sqrt{\phantom{x}}$$ always denotes the **principal (non-negative)** square root, the value of any square root expression is always greater than or equal to zero:

$$\sqrt{a} \geq 0 \quad \text{for all } a \geq 0$$

Therefore, no value of the variable can make a square root expression equal a negative number, and the equation has no solution. When this situation arises during Step 1 of the solving procedure — after isolating the radical — there is no need to continue with the remaining steps (squaring, solving, and checking). The absence of a solution is determined immediately.

Radical Equation with No Solution

Solve the radical equation $$\sqrt{9k - 2} + 1 = 0$$.

**Step 1 — Isolate the radical.** Subtract $$1$$ from both sides:

$$\sqrt{9k - 2} + 1 - 1 = 0 - 1$$

$$\sqrt{9k - 2} = -1$$

The isolated square root equals $$-1$$, which is a negative number. Since the radical sign always denotes the principal (non-negative) square root, $$\sqrt{9k - 2} \geq 0$$ for every allowable value of $$k$$. No real number substituted for $$k$$ can make the left side equal $$-1$$. Therefore, the equation has **no solution**.

This example demonstrates that when isolating the radical produces a negative value on the other side, the solving process stops immediately — there is no need to square both sides or perform any further algebraic steps.

Solving $$\sqrt{9k-2} + 1 = 0$$

Solve the radical equation $$\sqrt{p - 1} + 1 = p$$ using the four-step procedure. This equation is more complex than earlier examples because squaring both sides produces a **quadratic** equation that must be solved by factoring.

**Step 1 — Isolate the radical.** Subtract $$1$$ from both sides:

$$\sqrt{p - 1} + 1 - 1 = p - 1$$

$$\sqrt{p - 1} = p - 1$$

**Step 2 — Square both sides.** The left side simplifies via the Squaring Property. The right side is a binomial that must be expanded using the Binomial Squares Pattern $$(a - b)^2 = a^2 - 2ab + b^2$$:

$$(\sqrt{p - 1})^2 = (p - 1)^2$$

$$p - 1 = p^2 - 2p + 1$$

**Step 3 — Solve the new equation.** The result is a quadratic equation. Move all terms to one side to obtain standard form:

$$0 = p^2 - 3p + 2$$

Factor the right side: $$0 = (p - 1)(p - 2)$$. Apply the Zero Product Property:

$$p - 1 = 0 \quad \text{or} \quad p - 2 = 0$$

$$p = 1 \quad \text{or} \quad p = 2$$

**Step 4 — Check both answers** by substituting into the original equation:

For $$p = 1$$: $$\sqrt{1 - 1} + 1 = \sqrt{0} + 1 = 0 + 1 = 1$$, and the right side is $$1$$. Since $$1 = 1$$ is true, $$p = 1$$ is a solution. ✓

For $$p = 2$$: $$\sqrt{2 - 1} + 1 = \sqrt{1} + 1 = 1 + 1 = 2$$, and the right side is $$2$$. Since $$2 = 2$$ is true, $$p = 2$$ is a solution. ✓

The solutions are $$p = 1$$ and $$p = 2$$. Unlike earlier radical equation examples where squaring produced a linear equation, here the non-radical side $$(p - 1)$$ is a binomial that produces a trinomial $$p^2 - 2p + 1$$ when squared. Correctly applying the Binomial Squares Pattern — including the middle term $$-2p$$ — is essential. Omitting the middle term is a common error.

Solving $$\sqrt{p-1} + 1 = p$$

Solve the radical equation $$\sqrt{r + 4} - r + 2 = 0$$ using the four-step procedure. This equation produces a quadratic after squaring, and one of the two candidate solutions turns out to be extraneous.

**Step 1 — Isolate the radical.** Add $$r$$ and subtract $$2$$ from both sides (equivalently, move $$-r + 2$$ to the right side):

$$\sqrt{r + 4} = r - 2$$

**Step 2 — Square both sides.** The left side simplifies via the Squaring Property. The right side is a binomial that must be expanded using the Binomial Squares Pattern $$(a - b)^2 = a^2 - 2ab + b^2$$:

$$(\sqrt{r + 4})^2 = (r - 2)^2$$

$$r + 4 = r^2 - 4r + 4$$

**Step 3 — Solve the new equation.** Rearrange to standard form by moving all terms to one side:

$$0 = r^2 - 5r$$

Factor the right side: $$0 = r(r - 5)$$. Apply the Zero Product Property:

$$r = 0 \quad \text{or} \quad r - 5 = 0$$

$$r = 0 \quad \text{or} \quad r = 5$$

**Step 4 — Check both answers** by substituting into the original equation:

For $$r = 0$$: $$\sqrt{0 + 4} - 0 + 2 = \sqrt{4} + 2 = 2 + 2 = 4 \neq 0$$. This is **not** a solution — $$r = 0$$ is an extraneous solution.

For $$r = 5$$: $$\sqrt{5 + 4} - 5 + 2 = \sqrt{9} - 3 = 3 - 3 = 0$$. Since $$0 = 0$$ is true, $$r = 5$$ is a solution. ✓

The solution is $$r = 5$$. The value $$r = 0$$ is extraneous — it was introduced by the squaring step. When $$r = 0$$, isolating the radical gives $$\sqrt{4} = 0 - 2 = -2$$, which requires the principal square root to equal a negative number, an impossibility. This example reinforces that checking every candidate solution is essential, especially when squaring produces a quadratic with multiple solutions.

Solving $$\sqrt{r+4} - r + 2 = 0$$

Solve the radical equation $$3\sqrt{3x - 5} - 8 = 4$$ using the four-step procedure. This equation has a numerical coefficient multiplying the square root, so isolating the radical means isolating the entire product $$3\sqrt{3x - 5}$$ — and when both sides are squared, the coefficient is squared along with the radical.

**Step 1 — Isolate the radical.** Add $$8$$ to both sides to move the constant:

$$3\sqrt{3x - 5} = 12$$

**Step 2 — Square both sides.** Because the left side is $$3\sqrt{3x - 5}$$, squaring it means $$(3\sqrt{3x - 5})^2 = 3^2 \cdot (\sqrt{3x - 5})^2 = 9(3x - 5)$$. On the right, $$12^2 = 144$$:

$$(3\sqrt{3x - 5})^2 = 12^2$$

$$9(3x - 5) = 144$$

**Step 3 — Solve the new equation.** Distribute the $$9$$:

$$27x - 45 = 144$$

Add $$45$$ to both sides: $$27x = 189$$. Divide both sides by $$27$$: $$x = 7$$.

**Step 4 — Check.** Substitute $$x = 7$$ into the original equation:

$$3\sqrt{3(7) - 5} - 8 = 3\sqrt{21 - 5} - 8 = 3\sqrt{16} - 8 = 3(4) - 8 = 12 - 8 = 4$$

Since $$4 = 4$$ is true, $$x = 7$$ is confirmed as the solution. When a coefficient multiplies the radical, the coefficient must be squared along with the square root when applying the Squaring Property. Here, $$(3\sqrt{3x-5})^2 = 9(3x-5)$$, not $$3(3x-5)$$ — forgetting to square the coefficient is a common error.

Solving $$3\sqrt{3x-5} - 8 = 4$$

Solve the radical equation $$\sqrt{4z - 3} = \sqrt{3z + 2}$$, which has square root expressions on **both** sides of the equation.

**Step 1 — Isolate the radical.** Both radical terms are already isolated — each side of the equation consists of a single square root expression, so no rearrangement is needed.

**Step 2 — Square both sides.** Apply the Squaring Property to each side independently:

$$(\sqrt{4z - 3})^2 = (\sqrt{3z + 2})^2$$

$$4z - 3 = 3z + 2$$

**Step 3 — Solve the new equation.** Both radicals have been eliminated, leaving a linear equation. Subtract $$3z$$ from both sides: $$z - 3 = 2$$. Add $$3$$ to both sides: $$z = 5$$.

**Step 4 — Check.** Substitute $$z = 5$$ into the original equation:

$$\sqrt{4(5) - 3} = \sqrt{3(5) + 2}$$

$$\sqrt{17} = \sqrt{17}$$

Since $$\sqrt{17} = \sqrt{17}$$ is true, $$z = 5$$ is confirmed as the solution. When a radical equation has square roots on both sides, squaring eliminates both radicals simultaneously — the Squaring Property is applied to each side, and the result is a standard linear equation with no radicals remaining.

Solving $$\sqrt{4z-3} = \sqrt{3z+2}$$

In some radical equations, squaring both sides does **not** fully eliminate the radical — a variable may still remain inside a square root after the first squaring step. When this occurs, the solving procedure is not complete after a single application of Steps 1 and 2. Instead, the same two steps must be **repeated**: isolate the remaining radical on one side of the equation and then square both sides again. This second round of squaring removes the remaining radical, producing an equation that can be solved using standard algebraic techniques. The check step (Step 4) is still performed at the end, using the original equation.

Repeating the Isolate-and-Square Steps in Radical Equations

A safety inspector is using a formula that contains a square root to determine the maximum load for a warehouse shelf. To solve for the unknown variable, the inspector must follow a specific algebraic procedure. Arrange the following steps in the correct order to solve a radical equation.

A laboratory technician is following a standard four-step procedure to solve radical equations used in chemical concentration models. Match each step number with the corresponding action required in the procedure.

A quality control inspector is using a radical equation to determine the load-bearing capacity of a structural component. After the inspector has squared both sides of the equation and solved for the variable to find potential values, what is the final mandatory step in the procedure?

A safety technician is using a radical equation to calculate the maximum weight a support cable can hold. True or False: In the standard four-step procedure for solving such equations, the process of squaring both sides guarantees that any solution found will be a valid answer for the original radical formula.

A quality assurance analyst is solving a radical equation to calculate the tolerance of a mechanical component. After squaring both sides and solving for the variable, the analyst finds a potential value that does not satisfy the original equation. According to the standard four-step procedure, what is the specific mathematical term for this type of invalid solution?

Identifying Invalid Results in Radical Equations

A safety officer is solving a radical equation to calculate the load limit for a crane. To ensure the final answer is valid and not an extraneous solution, the officer must substitute the result back into the _________ equation.

Based on the standard four-step procedure for solving radical equations, identify the specific step that was omitted in the case below and provide the technical term for the type of invalid solution this step is intended to detect.

Diagnostic Audit of a Radical Equation Process

In your role as a Technical Documentation Lead, you are tasked with updating the company’s internal manual for standard mathematical protocols used in engineering reports. One section must explain how to handle formulas that involve a variable under a square root (radical equations). Write a brief essay outlining the standard four-step procedure for solving such equations. Your response must identify each step in the correct sequence and state the specific purpose of the final verification step in maintaining the integrity of the data.

Standard Operating Procedure for Radical Equation Analysis

A safety engineer is solving a radical equation to calculate the load-bearing capacity of a scaffolding system. After successfully isolating the radical expression on one side of the equation, the engineer prepares to square both sides. According to the standard four-step procedure, what is the specific purpose of this action?

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