Question

Solve the equation by completing the square. \(z(z+9)=1\)

Short answer

The solution of the equation \(z(z+9)=1\) by completing the square is \(z = 0.10977\) and \(z = -9.10977\)

Step-by-step solution

Make the equation in a standard quadratic form

First, change the equation into a standard quadratic equation form \(ax^2 + bx + c = 0\). Here, we get \(z^2 + 9z - 1 = 0\).

Derive 'b' in the quadratic equation

In the quadratic equation \(z^2 + 9z - 1 = 0\), 'b' is the coefficient of 'z', which is '9'.

Complete the square

To complete the square, the formula to calculate 'c' is \((b/2)^2\). So, the equation becomes \(z^2 + 9z + (9/2)^2 = 1 + (9/2)^2\) which simplifies to \(z^2 + 9z + 20.25 = 21.25\).

Write the equation in square form

Now, write the left side of the equation as a square: \((z + 9/2)^2 = 21.25\).

Solve for z

Take the square root on both sides to solve for 'z'. We get \(z + 9/2 = +- sqrt{21.25}\). Hence, \(z = -9/2 +- sqrt{21.25}\).

Simplify 'z' to obtain the final answer

Simplify the obtained values of 'z' to obtain the final answer. Solving it we get \(z = -4.5 +- 4.60977\). Hence, the solutions are \(z = 0.10977\) and \(z = -9.10977\).

Key concepts

Quadratic Equations

Quadratic equations are mathematical expressions that form a U-shaped curve when plotted on a graph, known as a parabola. These expressions have the general form: \[ ax^2 + bx + c = 0 \]where:

• \(a\), \(b\), and \(c\) are constants with \(a eq 0\)
• \(x\) is the variable that you solve for.

In our exercise, we start with an equation: \(z(z+9)=1\), which can be rewritten in the standard quadratic form as \(z^2 + 9z - 1 = 0\).

This form allows us to apply various methods for solving, such as factoring, using the quadratic formula, or completing the square. Parabolas can open upwards or downwards, depending on the sign of \(a\). In this particular equation, because the \(a\)-value is positive, an upward opening parabola is implied. Understanding these properties and patterns is essential to solve quadratic equations efficiently.

Solving Equations

Solving equations involves finding the value of the variable that satisfies the equation. For quadratic equations, there are multiple solving techniques such as factoring, using the quadratic formula, or completing the square. Each method involves step-by-step manipulation to isolate the variable.

In our exercise, we used the technique of completing the square. This is useful when the quadratic equation doesn't easily factor. It involves transforming the quadratic equation into a perfect square trinomial, making it simpler to solve. Once you have the equation in a form such as \((z + 9/2)^2 = 21.25\), it becomes straightforward to find the variable values by taking the square root of both sides.

Understanding the basic approach to solving equations helps in choosing the most effective method based on the problem at hand.

Square Root Method

The square root method is a key approach in solving quadratic equations that have been transformed into a square form. Once the trinomial is written as a perfect square, solving for the variable involves applying the square root to both sides of the equation.

Let's revisit our example:\[ (z + 9/2)^2 = 21.25 \]
By taking the square root of both sides, we get:
\[ z + 9/2 = \pm \sqrt{21.25} \]
This method yields two potential solutions since the equation can be positive or negative. It is crucial to isolate the variable completely by moving terms around appropriately. Once you obtain the numerical values, make sure to verify them in the original equation to confirm their validity.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations using mathematical operations to isolate the variable of interest. This might include adding, subtracting, multiplying, or dividing terms on both sides of the equation.

In our specific problem, we performed several algebraic steps:

• Rewriting \(z(z+9)=1\) into the standard form \(z^2 + 9z - 1 = 0\).
• Completing the square by adding \((9/2)^2\) to both sides to form a perfect square trinomial.
• Simplifying to isolate \(z\) using the square root method.

These steps require a keen understanding of how operations affect the equation's balance.

Algebraic manipulation is fundamental in problem-solving as it forms the building blocks for more complex mathematical techniques.