Problem 74 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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:\[d^2 - 8d = 9\]

Step 2: 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. \[\left(\frac{-8}{2}\right)^2 = 16\]

Step 3: 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:\[d^2 - 8d + 16 = 25\]

Step 4: Rewrite as a binomial square

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

Step 5: 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\[d - 4 = -5\]

Step 6: 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 \(d - 4 = -5\):\[d = -5 + 4\]\[d = -1\]

Key Concepts

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.