(week3A)Mathematics for Engineers Preliminary

A collection of flashcards covering key concepts in linear equations, simultaneous equations, quadratic equations, and trigonometric ratios as studied in Mathematics for Engineers.

What is a linear equation?

An equation of the form ax + b = 0, which contains only the first power of a single unknown.

How do you solve the linear equation 3x - 2 = 10 - x?

By rearranging the equation to 4x - 2 = 10, leading to x = 3.

What are simultaneous linear equations?

Equations containing two unknowns, which are solved together to find a common solution.

What is the elimination method in solving simultaneous equations?

It involves eliminating one variable to create a single equation in one variable.

What are the roots of a quadratic equation?

The values of x that satisfy the equation ax² + bx + c = 0.

What is a quadratic equation?

An equation that has the standard form ax² + bx + c = 0.

How do you solve a quadratic equation by factorization?

By expressing ax² + bx + c as (px + q)(rx + s) and solving the resulting linear equations.

What is the formula to find the roots of a quadratic equation?

x = (-b ± √(b² - 4ac)) / (2a).

What does the discriminant tell us about a quadratic equation?

It indicates the nature of the roots; if positive, there are two distinct roots; if zero, one repeated root; if negative, no real roots.

What are the values of the angle sum in a triangle?

The angle sum is 180 degrees or π radians.

What is the sine ratio in a right triangle?

sin(θ) = opposite/hypotenuse.

What is the cosine ratio in a right triangle?

cos(θ) = adjacent/hypotenuse.

What is the tangent ratio in a right triangle?

tan(θ) = opposite/adjacent.

What is the sine of 45 degrees (or π/4)?

1/√2.

What is the cosine of 30 degrees (or π/6)?

√3/2.

What is the tangent of 60 degrees (or π/3)?

√3.

How can you solve a quadratic equation by completing the square?

By rewriting the equation in the form (x + b/2)² = c.

What are the steps to find x in the equation x² + 6x + 7 = 0?

Complete the square to (x + 3)² - 2 = 0, leading to x = -3 ± √2.

What is the significance of the angle measurement in trigonometry?

Angles can be measured in degrees and radians, with radians being the preferred measurement in calculus.

How can the trigonometric ratio for an angle of 30 degrees (or π/6) be expressed?

sin(30°) = 1/2, cos(30°) = √3/2, tan(30°) = 1/√3.

What is the solution for the quadratic equation 6x² + 19x + 10 = 0?

The roots are x = -5/2 and x = -2/3.