Algebra Exam Review - Quadratics and Triangles

The distance between two points on a line segment is calculated using the distance formula: √((x2-x1)² + (y2-y1)²).

The perpendicular bisector of a line segment is a line that passes through the midpoint of the segment and is perpendicular to it.

Two equations can intersect at zero, one, or infinitely many points. You can find the solution by substitution or elimination.

A point is inside a circle if its distance from the center is less than the radius, outside if it's greater, and on the circle if it's equal.

To expand a factored form equation into standard form, multiply out the factors using the distributive property.

The vertex of a quadratic equation in factored form can be found by finding the average of the x-intercepts or by using completing the square.

The vertex of a quadratic equation in vertex form (y = a(x-h)² + k) is represented by the point (h, k).

Translations of quadratic equations involve shifting the graph up, down, left, or right. To translate, add or subtract constants to the x or y terms in the equation.

The minimum or maximum height of a quadratic equation is found at the vertex. If the coefficient of the quadratic term is positive, the parabola opens upwards and the vertex is a minimum. If it's negative, the parabola opens downwards and the vertex is a maximum.

Solve right-angled triangles using SOHCAHTOA (Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent).

Solve non-right angled triangles using the sine and cosine laws. The Sine Law relates sides and angles: a/sinA = b/sin B = c/sinC. The Cosine Law relates sides and angles: a² = b² + c² - 2bc cosA.

Expand and simplify a factored form equation into standard form by multiplying out the factors (using distributive property) and combining like terms.

Factoring standard equations into factored form requires finding two binomials whose product is the original equation. Methods include common factoring, simple trinomial factoring, decomposition, and difference of squares.

Complete the square to transform a standard form equation into vertex form by rearranging terms, completing the square by adding a constant to both sides, and factoring.

Complete the square to convert a standard form quadratic equation into vertex form (y = a(x-h)² + k). This vertex form reveals the vertex of the parabola.

Graphing a quadratic equation in vertex form involves plotting the vertex and using the leading coefficient to determine the direction and shape of the parabola.

Word problems involving quadratic relationships often require setting up equations and solving for unknowns. For example, a fence or picture frame problem involves calculating dimensions for a specific area or perimeter.

Quadratic equations can be used to model scenarios involving projectiles or other objects in motion. Solving for the zeros (roots or x-intercepts) tells us when the object is at ground level.

The cosine law is used to find missing side lengths and angles in any triangle, not just right-angled ones. It relates the sides and angle of a triangle: a² = b² + c² - 2bc cosA.

Trigonometric ratios (SOHCAHTOA) are applied to right-angled triangles to solve for side lengths and angles. It involves relating angles to ratios of sides.