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Chapter 10: Problem 81

Solve by completing the square. \(a^{2}-10 a=-5\)

Short Answer

Expert verified
The solutions are \( a = 5 + 2\sqrt{5} \) and \( a = 5 - 2\sqrt{5} \).

Step by step solution

01

Move the constant term to the right side

Start by adding 5 to both sides to isolate the quadratic and linear terms on the left side: \[ a^{2} - 10a = -5 \] Adding 5 to both sides:\[ a^{2} - 10a + 5 \rightarrow 0 \] So, the equation is now: \[ a^{2} - 10a + 5 = 0 \]
02

Add and subtract a specific value to complete the square

To complete the square, take half the coefficient of the linear term (-10), square it, and add it to both sides.Half of -10 is -5, and (-5) squared is 25. Add and subtract 25: \[ a^{2} - 10a + 25 - 25 + 5 = 0 \]
03

Rearrange and simplify

Combine and simplify the terms inside the parentheses:\[ (a - 5)^2 - 25 + 5 = 0 \]Simplify further: \[ (a - 5)^2 - 20 = 0 \]
04

Isolate the square term

Add 20 to both sides to isolate the squared term:\[ (a - 5)^2 = 20 \]
05

Solve for a

Take the square root of both sides to solve for a: \[ a - 5 = \pm\sqrt{20} \]Remember that \sqrt{20} simplifies to 2\sqrt{5}.So, \[ a - 5 = \pm 2\sqrt{5} \]Finally, add 5 to both sides to solve for a:\[ a = 5 \pm 2\sqrt{5} \]

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Key Concepts

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

Quadratic Equations
Quadratic equations are equations of the form 
ax^2 + bx + c = 0
where a, b, and c are constants.
The solutions to a quadratic equation are the values of x that make the equation true.
Examples of quadratic equations are 2x^2 + 3x - 5 = 0 and x^2 - 4x + 4 = 0.
There are various methods to solve quadratic equations:
  • Factoring
  • Using the quadratic formula
  • Graphing
  • Completing the square

In our exercise, we use the completing the square method.
Solving Equations
Solving equations involves finding the value of the variable(s) that make the equation true.
There are different types of equations, like linear, quadratic, and polynomial.
In solving quadratic equations, the goal is to isolate the variable.
For our exercise, we follow specific steps:
  • Move constant terms to one side
  • Complete the square
  • Isolate the variable

By moving terms around, combining like terms, and manipulating the equation, we solve for the variable.
Each step focuses on making the equation simpler and isolating the unknown variable.
Algebraic Manipulation
Algebraic manipulation involves rearranging equations to make them easier to solve.
This includes combining like terms, using arithmetic operations, and applying algebraic identities.
In the context of completing the square:
  • Move constants to isolate the quadratic and linear terms
  • Add and subtract specific values to form a perfect square trinomial
  • Factor the trinomial
  • Isolate the squared term
These steps require understanding how to balance equations and manipulate them while maintaining equality.

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Most popular questions from this chapter

Rafi is designing a rectangular playground to have an area of 320 square feet. He wants one side of the playground to be four feet longer than the other side. Solve the equation \(p^{2}+4 p=320\) for \(p,\) the length of one side of the playground. What is the length of the other side?

Solve by using the Quadratic Formula. \(\frac{1}{9} c^{2}+\frac{2}{3} c=3\)

In the following exercises, find (a) the axis of symmetry and (b) the vertex. $$ y=-2 x^{2}-8 x-3 $$

In the following exercises, graph by using intercepts, the vertex, and the axis of symmetry. $$ y=x^{2}+6 x+5 $$

Solve by using the Quadratic Formula. \(2 x^{2}+12 x-3=0\)
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