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Chapter 10: Problem 85

Solve by completing the square. \(v^{2}=9 v+2\)

Short Answer

Expert verified
v= -4 or v+5

Step by step solution

01

Move constants to one side

Rewrite the equation to move the constant term to the other side. Subtract 9v from both sides:
02

Form a perfect square trinomial

For completing the square, find the value that completes the square. Half the coefficient of v (which is -9), then square it:
03

Add and subtract the square value

Add and subtract this value inside the equation:
04

Factor the perfect square trinomial

Factorize the perfect square trinomial into
05

Solve for v

Solve the simplified equation by isolating v, first take the square root of both sides then solve for the two possible values of v that satisfy the equation v = -4 or v-5

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

solving quadratic equations
Solving quadratic equations is a fundamental aspect of algebra. Quadratic equations are in the form of ax^2 + bx + c = 0. To solve these equations, there are several methods you can use:
  • Factorization
  • Using the quadratic formula
  • Completing the square
Completing the square is a method where you rewrite the quadratic equation in a way that makes it easier to solve. For example, consider the equation v^2 = 9v + 2.
In our case, the first step is to move all the terms involving variables to one side of the equation and constants to the other side. So it becomes: v^2 - 9v = 2. The goal is to transform this into a perfect square trinomial because it makes the equation easier to factor.
perfect square trinomial
A perfect square trinomial is a special form of a quadratic equation where you have something like (x+d)^2, which expands to x^2 + 2dx + d^2. To convert a regular quadratic equation into a perfect square trinomial:
  • Take the coefficient of the linear term (the term with v).
  • Divide this coefficient by 2.
  • Square the result of this division.
Going back to our example, the coefficient of v is -9. Half of -9 is -9/2. Squaring (-9/2) gives us 81/4.
Once you have this value, you add and subtract it in the equation to keep it balanced. So our equation becomes: v^2 - 9v + 81/4 - 81/4 = 2. This is allowed because adding and subtracting the same value doesn't change the equation. Then you group the perfect square trinomial: v^2 - 9v + 81/4 and treat -81/4 as a constant on the other side.
factoring
Factoring involves rewriting an equation as a product of simpler expressions that can be solved more easily. For the equation we have formed, v^2 - 9v + 81/4, we need to express it as a squared term:
This can be rewritten as (v - 9/2)^2. Then, rewrite the right side of the equation combining constants: 2 + 81/4. Simplify this to get 89/4. Now, the equation looks like (v - 9/2)^2 = 89/4. Finally, solve for v by taking the square root of both sides: v - 9/2 = 卤鈭�(89/4), which simplifies to: v - 9/2 = 卤鈭�(89)/2.
Solving this gives us two possible solutions, v = (9 + 鈭�89)/2 and v = (9 - 鈭�89)/2. This is the final solution to our original quadratic equation.

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Most popular questions from this chapter

In the following exercises, graph by using intercepts, the vertex, and the axis of symmetry. $$ y=-16 x^{2}+24 x-9 $$

An architect is designing a hotel lobby. She wants to have a triangular window looking out to an atrium, with the width of the window 6 feet more than the height. Due to energy restrictions, the area of the window must be 140 square feet. Solve the equation \(\frac{1}{2} h^{2}+3 h=140\) for \(h,\) the height of the window.

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