Regents Algebra & Geometry 1

Front 1

How do you solve inequalities?

Back 1

In equalities are just like equations with one exception -- when an inequality is multiplied or divided by a negative number, the sign of the inequality flips: > becomes <, < becomes >, etc.

Front 2

How do you multiply 2 binomials together? For example, what is (x+a)(x+b)?

Back 2

Use the FOIL method -- which stands for First Outside Inside Last. For example, (x+a)(x+b) = x² + bx + ax + ab.

Front 3

How do you factor a quadratic of the form x²+bx+c, such as x²+5x+6?

Back 3

An equation of this form, such as x²+5x+6 will factor into the form (x+?)(x+?) where the missing terms are factors of 6 that add up to 5. To find these terms, just write out the factors of 6 and pick the pair that adds up to 5. These are 3 and 2, so x²+5x+6=(x+3)(x+2).

Front 4

How do you use factoring to solve a quadratic equation such as x²+5x+6=0?

Back 4

1. Factor the expression and set the product of factors equal to zero.
Since x²+5x+6 can be factored into (x+3)(x+2), you can set (x+3)(x+2)=0.
2. A product can only equal zero when one of its factors is qero. Therefore, either x+3=0 or x+2=0.
3. If x+3=0, x=-3; if x+2=0, x=-2. Thus, x=-2 or -3.

Front 5

How do you factor x²-4x-21?

Back 5

x²-4x-21 = (x+?)(x+?) where the missing terms are factors of -21 that add up to -4. The factors of -21 are 1 & -21, 3 & -7, 7 & -3, and 21 & -1. The only pair of those that add up to -4 are 3 & -7. Therefore, (x+3)(x-7) is the answer.

Front 6

Solve 2x²+3x-5 = 0.

Back 6

1. Factor 2x²+3x-5. The 2 factors will be of the form (2x+?)(x+?) where the missing terms are factors of -5 that will make the the binomials multiply to 2x²+3x-5 using the FOIL method. Using experimentation, you find that (2x+5)(x-1) works.
2. If (2x+5)(x-1)=0, one of the 2 factors must =0. On the Regents, they will only ask you to find the integral solution. In this case, if x-1=0, x=1. (Note: the other solution is 5/2).

Front 7

What are complementary angles?

Back 7

Angles that add up 90° (a right angle).

Front 8

What are supplementary angles?

Back 8

Supplementary angles are angles that add up to 180° (a straight angle).

Front 9

When 2 lines intersect, 1) what is true about 2 adjacent angles that are formed and 2) what is true about 2 opposite angles that are formed (i.e. "vertical angles")?

Back 9

When 2 lines intersect, 1) 2 adjacent angles are supplementary (i.e. add up to 180 and 2) 2 opposite angles, which are called vertical angles, are equal in measure.

Front 10

When parallel lines are intersected by a third line (called a transversal), 2 kinds of angles are formed -- big and small. What is true about these angles?

Back 10

1. All of the big angles are equal.
2. All of the small angles are equal.
3. Any big angle plus any small angle equals 180°.

Front 11

What is the sum of the interior angles of a triangle?

Back 11

The sum of the interior angles of a triangle is 180°.

Front 12

What do you call a triangle with 2 equal sides?

Back 12

A triangle with 2 equal sides is called an isosceles triangle. Also, an isosceles triangle has 2 equal angles, opposite the 2 equal sides.

Front 13

What do you call a triangle with 3 sides of different lengths?

Back 13

A triangle with 3 sides of different lengths is called a scalene triangle. In a scalene triangle, all 3 angles are of different measure.

Front 14

What do you call a triangle with 3 equal sides?

Back 14

An equilateral triangle.

Front 15

What does each angle of an equilateral triangle measure?

Back 15

Each angle of an equilateral triangle measures 60°.

Front 16

What is a right triangle?

Back 16

A right triangle is a triangle with a right angle (90°).

Front 17

In a right triangle, what are the 3 sides called?

Back 17

The 2 shorter (perpendicular) sides are called the legs while the longest side, opposite the right angle, is called the hypoteneuse.

Front 18

How are the sides of a right triangle related?

Back 18

The sides of a right triangle are related by the Pythagorean Theorem: a²+b²=c² where a and b are the legs and c the hypoteneuse.

Front 19

What do you call right triangles whose sides have integer lengths?

Back 19

Right triangles whose sides have integer lengths are called Pythagorean triplets.

Front 20

Name some Pythagorean triplets (i.e. right triangles whose sides have integer lengths).

Back 20

The most common Pythagorean triplet is a 3-4-5 triangle. Other common triplets are multiples of 3-4-5 (such as 6-8-10), 5-12-13 and 7-24-25.

Front 21

What is the formula for the area of a triangle?

Back 21

The area of a triangle A=½bh where b is the base and h is the height.

Front 22

What is the measure of an exterior angle of a triangle?

Back 22

The measure of an exterior angle equals the sum of the measures of the 2 nonadjacent interior angles.

Front 23

What is the relationship between the 3 lengths of any triangle?

Back 23

The length of any side of a triangle must be greater than the difference between the lengths of the other 2 sides and less than their sum. For example, if 2 sides of a triangle are 10 and 15, the 3rd side must be >5 and <25.

Front 24

In a 30-60-90 triangle, if the shorter leg is 1, what are the other 2 sides?

Back 24

If the shorter leg is 1, the longer side is the square root of 3 and the hypoteneuse is 2.

Front 25

In a 45-45-90 triangle, if the legs are 1, what is the hypoteneuse?

Back 25

In a 45-45-90 triangle with the legs = 1, the hypoteneuse is the square root of 2.

Front 26

If the altitude in a right triangle is drawn, what is true about the 2 triangles that are created?

Back 26

The 2 triangles are similar to each other and also similar to the large triangle; that is, the lengths of their sides are proportional.

Front 27

How can you find the largest angle of a triangle?

Back 27

The largest angle of a triangle is the one opposite the largest side. Also, the smallest angle is the one opposite the smallest side.