# Regents Algebra &amp; Geometry 1

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How do you solve inequalities?
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.
How do you multiply 2 binomials together? For example, what is (x+a)(x+b)?
Use the FOIL method -- which stands for First Outside Inside Last. For example, (x+a)(x+b) = x² + bx + ax + ab.
How do you factor a quadratic of the form x²+bx+c, such as x²+5x+6?
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).
How do you use factoring to solve a quadratic equation such as x²+5x+6=0?
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.
How do you factor x²-4x-21?
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.
Solve 2x²+3x-5 = 0.
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).
What are complementary angles?
Angles that add up 90° (a right angle).
What are supplementary angles?
Supplementary angles are angles that add up to 180° (a straight angle).
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")?
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.
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?
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°.
What is the sum of the interior angles of a triangle?
The sum of the interior angles of a triangle is 180°.
What do you call a triangle with 2 equal sides?
A triangle with 2 equal sides is called an isosceles triangle. Also, an isosceles triangle has 2 equal angles, opposite the 2 equal sides.
What do you call a triangle with 3 sides of different lengths?
A triangle with 3 sides of different lengths is called a scalene triangle. In a scalene triangle, all 3 angles are of different measure.
What do you call a triangle with 3 equal sides?
An equilateral triangle.
What does each angle of an equilateral triangle measure?
Each angle of an equilateral triangle measures 60°.
What is a right triangle?
A right triangle is a triangle with a right angle (90°).
In a right triangle, what are the 3 sides called?
The 2 shorter (perpendicular) sides are called the legs while the longest side, opposite the right angle, is called the hypoteneuse.
How are the sides of a right triangle related?
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.
What do you call right triangles whose sides have integer lengths?
Right triangles whose sides have integer lengths are called Pythagorean triplets.
Name some Pythagorean triplets (i.e. right triangles whose sides have integer lengths).
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.
What is the formula for the area of a triangle?
The area of a triangle A=½bh where b is the base and h is the height.
What is the measure of an exterior angle of a triangle?
The measure of an exterior angle equals the sum of the measures of the 2 nonadjacent interior angles.
What is the relationship between the 3 lengths of any triangle?
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.
In a 30-60-90 triangle, if the shorter leg is 1, what are the other 2 sides?
If the shorter leg is 1, the longer side is the square root of 3 and the hypoteneuse is 2.
In a 45-45-90 triangle, if the legs are 1, what is the hypoteneuse?
In a 45-45-90 triangle with the legs = 1, the hypoteneuse is the square root of 2.
If the altitude in a right triangle is drawn, what is true about the 2 triangles that are created?
The 2 triangles are similar to each other and also similar to the large triangle; that is, the lengths of their sides are proportional.
How can you find the largest angle of a triangle?
The largest angle of a triangle is the one opposite the largest side. Also, the smallest angle is the one opposite the smallest side.

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