Question

What is a logical first step in solving the equation?

(a)${\left(x+5\right)}^2=64$

(b)${\left(x+5\right)}^2+5=64$

(c)$x^2+x=2$

Short answer

1. The first step to solve the given equation is to take square root on each side of the equation.
2. The first step to solve the given equation is to add (-5) on each side of the equation.
3. The first step to solve the given equation is to take x common from LHS of the equation.

Step-by-step solution

Part a. Step 1. Apply the concept of equivalent equations.

Two equations with exactly the same solutions are called equivalent equations.

Part a. Step 2. Apply the properties of equality.

• Adding the same quantity to both sides of an equation gives an equivalent equation.
• Multiplying both sides of an equation by the same nonzero quantity gives an equivalent equation.
• Squaring/Taking square roots on both sides of an equation gives an equivalent equation.

Part a. Step 3. Analyze the given expression.

Given

$\begin{array}{l}{\left(x+5\right)}^2=64\\ \Rightarrow {\left(x+5\right)}^2=8^2\end{array}$

To solve this, we first take square root each side to eliminate the square.

$\begin{array}{l}{\left(x+5\right)}^2=64\\ \Rightarrow {\left(x+5\right)}^2=8^2\\ \Rightarrow x+5=8\end{array}$

So, the first step to solve the given equation is to take square root on each side of the equation.

Part b. Step 1. Apply the concept of equivalent equations.

Two equations with exactly the same solutions are called equivalent equations.

Part b. Step 2. Apply the properties of equality.

• Adding the same quantity to both sides of an equation gives an equivalent equation.
• Multiplying both sides of an equation by the same nonzero quantity gives an equivalent equation.
• Squaring/Taking square roots on both sides of an equation gives an equivalent equation.

Part b. Step 3. Analyze the given expression.

Given

${\left(x+5\right)}^2+5=64$

To solve this, we first add -5 on each side of the equation to eliminate 5 from LHS.

$\begin{array}{l}{\left(x+5\right)}^2+5=64\\ \Rightarrow {\left(x+5\right)}^2+5-5=64-5\\ {\left(x+5\right)}^2=59\end{array}$

So, the first step to solve the given equation is to add -5 on each side of the equation.

Part c. Step 1. Apply the concept of equivalent equations.

Two equations with exactly the same solutions are called equivalent equations.

Part c. Step 2. Apply the properties of equality.

• Adding the same quantity to both sides of an equation gives an equivalent equation.
• Multiplying both sides of an equation by the same nonzero quantity gives an equivalent equation.
• Squaring/Taking square roots on both sides of an equation gives an equivalent equation.

Part c. Step 3. Analyze the given expression.

Given

$x^2+x=2$

To solve this, we first take x common from LHS.

$\begin{array}{l}{x}^2+x=2\\ \Rightarrow x\left(x+1\right)=1\left(1+1\right)\end{array}$

So, the first step to solve the given equation is to take x common from LHS of the equation.