Try the Free Math Solver or Scroll down to Tutorials!












Please use this form if you would like
to have this math solver on your website,
free of charge.

The Square Root Property and Completing the Square

10.1 The Square Root Property and Completing the Square

An quadratic equation is written in standard form if it is written as ax2+bx+c =
0 with (a ≠ 0).

If x2 = k then OR .

Examples: Solve each equation.

Steps to Completing the Square for an equation of the form x2 + bx + c = 0:

1. But the terms with variables on one side and the constant on the other side.
2. Find .

3. Add the square to both sides.

4. Factor the perfect square trinomial on the LHS.

5. Solve the equation using the methods discussed earlier.

Let's practice step 2:

Find the constant that should be added to get a perfect square trinomial.

Examples: Solve each equation by completing the square:

10.2 The Quadratic Formula

The solutions to ax2 + bx + c = 0(a ≠ 0) are given by


1. Make sure you put the equation in standard form FIRST!

2. Be very careful when simplifying your answers.

Examples: Solve each equation using the quadratic formula.

What is wrong with the following "solution" of 5x2 - 5x + 1 = 0?

10.3 Equations in Quadratic Form

Solve each equation:

Solve each problem:

1. In 1.75 hr, Ken rows his boat 5 mi upriver and comes back. The speed of the
current is 3 mph. How fast does Ken row?

2. Two chefs are preparing a banquet. One chef could prepare the banquet in 2 hr
less time than the other. Together they complete the job in 5 hr. How long will it
take the faster chef working alone?

Solve each equation. Be sure to check your answers.

Summary of Ways to solve ax2 + bx + c = 0:

Method Advantages Disadvantages
Factoring This is usually the
fastest method.
Not all polynomials are
factorable; some factorable
polynomials are hard to factor.
Square root
This is the simplest
method for solving
equations of the form
(ax + b)2 = c.
Few equations are given in
this form.
the square
This method can
always be used,
although most people
prefer the quadratic
It requires more steps than
other methods.
This method can
always be used.
It is more difficult than
factoring because of the
square root, although
calculators can simplify its use.

10.4 Formulas and Applications

Solve for the given variable:

Word Problems:

Example 3 on pg. 592: Two cars left an intersection at the same time, one heading
due north, the other due west. Some time later, they are exactly 100 mi apart. The
car headed north had gone 20 mi further than the car headed west. How far had each
car traveled?

#26 on pg. 596: A piece of sheet metal is 2 in. longer than it is wide. A square
piece 3 in. on a side is cut from each corner. The sides are then turned up to form an
uncovered box of volume 765 in.3. Find the dimensions of the original piece of metal.

#5 on pg. 594: A ball is propelled vertically upward from the ground. Its distance
in feet from the ground at t seconds is s(t) = -16t2 + 64t. At what times will the
ball be 32 ft from the ground?

10.5 Graphs of Quadratic Functions

The graph of a quadratic equation is called a parabola.

Things to know about graphs of quadratic equations:

1. The graph of f(x) = a(x - h)2 + k, a ≠ 0 is a parabola with vertex (h, k) and
vertical line x = h as axis.

2. The graph opens up if a is positive and down if a is negative.

3. The graph is winder than that of f(x) = x2 if 0 < lal < 1. The graph is narrower
than that of f(x) = x2 if lal> 1.

Let's practice determining the vertex of a parabola and whether it opens upward or

Graph the following. Give the vertex, axis, domain and range.