# Tag Archives: quadratic equation

## Tangent Circles

Consider the circles that have radii and are tangent to the line at the point . Find the sum of the coordinates of the centers of the circles. Source: NCTM Mathematics Teacher, August 2006 Solution For the two circles to … Continue reading

## Product of Consecutive Integers

How many integer pairs satisfy the equation ? Source: NCTM Mathematics Teacher, September 2006 Solution We construct products of positive consecutive integers and see which ones if any can be expressed as a power of . is the only such … Continue reading

## Length of Side of Square

If the numerical value of the area of a square plus two times the numerical value of its perimeter is equal to , what is the length of one side of the square? Source: NCTM Mathematics Teacher, September 2006 Solution … Continue reading

## Nested Square Roots 2

Find the value of Source: NCTM Mathematics Teacher, October 2006 Solution Let By substitution Squaring both sides Using the quadratic formula to solve for is an extraneous solution because the expression cannot be negative. Answer:

## Complicated Quadratic Root

What is the (simplified) value for which the larger root of is ? Source: NCTM Mathematics Teacher, October 2006 Solution We are going to take advantage of the formulas and repeatedly in the process of simplifying the complicated root. First, we multiply the … Continue reading

## Vertices of a Square

If and are the coordinates of two opposite vertices of a square, what are the coordinates of the other two vertices? Source: NCTM Mathematics Teacher, October 2006 Solution Let and be the two given opposite vertices of the square. The … Continue reading

## Points on Circle

Find all points that have an -coordinate twice the -coordinate and that lie on a circle of radius with center at . Source: NCTM Mathematics Teacher, January 2006 SOLUTION Let for some real number . Equation of circle or or … Continue reading

## Value of Expression

Given that , determine the value of . Source: NCTM Mathematics Teacher, January 2006 SOLUTION implies that . Answer: Alternative Solution

## Circles Within Square

The two circles in the figure are congruent and have their centers on the line . They are tangent to each other and to the square . The radius of the circles is . Calculate the area of the square … Continue reading

## Inscribed Squares

Two squares are inscribed in a semicircle, as shown. The area of the smaller square is . What is the area of the larger square? Source: NCTM Mathematics Teacher, December 2005 SOLUTION is the center of the circle and also … Continue reading