Math Problem Statement
Solution
Let's go through each problem in the image step-by-step:
Problem 1: Equations of Lines and Circle
(a) Write the equation of the line passing through points and .
- Calculate the slope :
- Use point-slope form with point : Expand to find the line equation.
(b) Write the equation of the line passing through and perpendicular to .
- Find the slope of by rewriting it in slope-intercept form .
- The perpendicular slope is the negative reciprocal.
- Use point-slope form with point to find the line equation.
(c) Find the area and perimeter of the circle with equation .
- Complete the square to rewrite in standard form .
- Calculate the radius , then:
- Area =
- Perimeter (circumference) =
Problem 2: Solving Systems and Inequalities
(a) Solve the system of equations:
x = 2 - y^2 \\ 2 = 3x + y \end{cases}$$ Substitute $$x = 2 - y^2$$ into the second equation and solve for $$y$$, then find $$x$$. **(b)** Solve the inequality $$x^2 + x - 6 > 0$$. 1. Factor the quadratic as $$(x + 3)(x - 2) > 0$$. 2. Determine intervals where the inequality holds. --- ### Problem 3: Polynomials and Binomial Theorem **(a)** Given $$(x - 2)$$ as a factor of $$x^3 + x^2 - 2x - 8$$, find the other factor. 1. Perform synthetic or polynomial division to divide $$x^3 + x^2 - 2x - 8$$ by $$x - 2$$. **(b)** Find the term with $$x^7$$ in the expansion of $$(3 + x)^{12}$$. 1. Use the binomial term formula $$\binom{n}{k} a^{n-k} b^k$$. 2. Set up for the term where $$k = 7$$. **(c)** Find the fourth term in $$(3 + x)^{12}$$ using the binomial theorem. 1. Use the formula for the binomial expansion term $$T_{k+1} = \binom{n}{k} a^{n-k} b^k$$ with $$n = 12$$ and $$k = 3$$. --- Would you like detailed solutions for each part, or have specific questions? --- ### Related Questions 1. How do you find the equation of a line through two given points? 2. How do you find a perpendicular line to a given line? 3. What are the steps for completing the square in a circle's equation? 4. How do you solve a system of nonlinear equations? 5. What is the general approach to solving polynomial inequalities? ### Tip For problems involving the binomial theorem, remember the general term formula, $$T_{k+1} = \binom{n}{k} a^{n-k} b^k$$, to quickly find any specific term in the expansion.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Algebra
Coordinate Geometry
Polynomial Division
Systems of Equations
Inequalities
Binomial Theorem
Formulas
Point-slope form of a line: y - y1 = m(x - x1)
Slope formula: m = (y2 - y1) / (x2 - x1)
Circle equation: (x - h)^2 + (y - k)^2 = r^2
Binomial term formula: T(k+1) = C(n, k) * a^(n-k) * b^k
Theorems
Binomial Theorem
Properties of Polynomial Division
Suitable Grade Level
Grades 10-12
Related Recommendation
Math Problem Solutions: Fractions, Polynomial Division, Simultaneous Equations
Comprehensive Model Paper: Algebra, Geometry, and Matrix Operations
Mathematical Problems: Factorization, Linear Functions, and Geometry Solutions
Math Problems for 9th Grade: Algebra, Geometry, and More
Advanced Math Problems in Algebra, Geometry, and Polynomials for Grade 8-10 Students