Regents Algebra & 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?
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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.
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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?
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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.