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Chapter 10: Problem 74

Solve by completing the square. \(d^{2}-8 d=9\)

Short Answer

Expert verified
The solutions are \(d = 9\) and \(d = -1\).

Step by step solution

01

Move the constant term to the other side

First, subtract 9 from both sides of the equation to move the constant term to the other side:\[d^2 - 8d - 9 = 0\]This simplifies to:\[d^2 - 8d = 9\]
02

Prepare to complete the square

Identify the coefficient of the linear term (the term involving \(d\)). In this case, it is -8. Take half of this coefficient and then square it. \[\left(\frac{-8}{2}\right)^2 = 16\]
03

Add and subtract the squared term

Add and subtract 16 inside the equation to balance it. This means adding 16 to both sides of the equation:\[d^2 - 8d + 16 - 16 = 9 + 16\]Which simplifies to:\[d^2 - 8d + 16 = 25\]
04

Rewrite as a binomial square

This allows you to rewrite the left side as a perfect square trinomial. The equation becomes:\[(d - 4)^2 = 25\]
05

Solve for the variable by taking the square root

Take the square root of both sides to solve for \(d\) :\[d - 4 = \pm 5\]This yields two solutions:\[d - 4 = 5\] or\[d - 4 = -5\]
06

Solve for the variable

Finally, add 4 to both sides of each equation to solve for \(d\) :For \(d - 4 = 5\):\[d = 5 + 4\]\[d = 9\]For \(d - 4 = -5\):\[d = -5 + 4\]\[d = -1\]

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Key Concepts

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

Quadratic Equations
A quadratic equation is a type of polynomial equation of the form:


































Binomial Square
A binomial square is a special expansion of the form:







































Solving Equations
Solving equations involves finding the values of variables that satisfy the given equation.















































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Most popular questions from this chapter

In the following exercises, find the maximum or minimum value. $$ y=-4 x^{2}+12 x-5 $$

Make up a problem involving the product of two consecutive odd integers. Start by choosing two consecutive odd integers. (a) What are your integers? (b) What is the product of your integers? (c) Solve the equation \(n(n+2)=p,\) where \(p\) is the product you found in part (b). (1) Did you get the numbers you started with?

In the following exercises, solve. Round answers to the nearest tenth. A stone is thrown vertically upward from a platform that is 20 feet high at a rate of \(160 \mathrm{ft} / \mathrm{sec}\). Use the quadratic equation \(h=-16 t^{2}+160 t+20\) to find how long it will take the stone to reach its maximum height, and then find the maximum height.

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

Recognize the Graph of a Quadratic Equation in Two Variables. $$ y=-x^{2}+1 $$
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