Quadratic equations Questions and Answers

The cubic polynomial equation x³ + bx² + cx + d = 0 has three roots X₁, X2, and X3. By expanding the product (x-x₁)(x - x₂)(x - x3), show that a) i) b = (x₁ + x₂ + x3); ii) c = x₁x₂ + x₁ x3 + x2X3; iii) d = -X₁X2X3. It is given that b = -9 and c = 45 for parts b) and c) below. b) i) In the case that the three roots x₁, x2, and x3 form an arithmetic sequence, show that one of the roots is 3. ii) Hence, determine the value of d. c) In another case the three roots form a geometric sequence. Determine the value of d.

Write an equation representing the fact that the product of two consecutive even integers is 48. Use x to represent the smaller integer. (b) Solve the equation from part (a) to find the two integers. (a) The equation is x (x + 2) (b) Find the pair(s) of consecutive even integers. If there is more than one pair, use the "or" button.

Dilbert recently hired a roofer to do some necessary work. On the final bill, Dilbert was charged a total of $1160. $480 was listed for parts and the rest for labor. If the hourly rate for labor was $68, how many hours of labor was needed to complete the job? (A) First write an equation you can use to answer this question. Use a as your variable. The equation is (B) Solve your equation in part (A) to find the number of labor hours needed to do the job. Answer: The number of labor hours was

The path of a shot put released at an angle of 35° can be modeled by y=-0.01x² +0.7x + 6, where x is the horizontal distance (in feet) and y is the vertical distance (in feet). Find and interpret the coordinates of the vertex. The coordinates of the vertex are (.. When the shot put is at its highest point, it is feet from its starting point and feet off the ground.

The cost for ground shipping with FedEx and UPS varies. UPS charges $8.25 initially, and then $0.20 per pound. FedEx initially charges $9.50 and then $0.16 per pound. Write the equation to find the number of pounds when the costs of shipping with UPS and FedEx are the same. (Let the number of additional pounds = x.)

Solve the equation by factoring. = 6x + 27 Rewrite the equation in factored form. = 0 (Factor completely.) The solution set is. (Use a comma to separate answers as needed. Type each solution only once.)

The sum of two polynomials is shown below. Polynomial p: 3x + 6 Polynomial q 42³ - 8x p+q = (3x+6) + (4x³ - 8x) = 4x³ - 5x + 6 How does the sum 4x³ 5x + 6 demonstrate the closure of polynomials under addition? Explain your answer.

Solve the equation by factoring. 3x(x-2)=8x² - 10x Rewrite the equation in factored form. =0 (Factor completely.) The solution set is. (Use a comma to separate answers as needed. Type each solution only once.)

Write the polynomial f(x) that meets the given conditions. Answers may vary. Degree 3 polynomial with zeros of -4, 5i, and -5i.

Find any intercepts. y = x² + 7x - 8 y-intercept: x-intercepts: (x, y) = (x, y) = (x, y) = (smaller x-value) (larger x-value)

Solve the following using substitution. If one solution, type that point in for both points. If no solution, type in "NS". y = x2 - 5x + 7 y = 2x + 1 Solution 1: x = ;y= Solution 2: x = ;y=

Given f (x) = x² + 2x - 7 and g(x) = 2x + 2, determine the x values for which f(x) = g(x). Round your answers to the nearest hundredth. x= -2.24 or x = 2.24 x= -5.61 or x = 1.61 x=3 or x = -3 No solutions because f(x) does not intersect g(x).

A motorboat maintained a constant speed of 12 miles per hour relative to the water in going 23 miles upstream and then returning. The total time for the trip was 24.0 hours. Use this information to find the speed of the current. The speed of the current is Y

Solve for v. 3/5v-25 +1=-1/v-5 If there is more than one solution, separate them with commas. If there is no solution, click on "No solution".

Use function notation to rewrite the function using the given function y = 8x² +7x+5, function name f Choose the correct function below. A. 8f2 +7f=y-5 B. y=8f² +7f+5 C. f(y) = 8x² + 7x+5 D. f(x)= 8x2 +7x+5

Use the long division algorithm to determine the quotient in problems 1-6. 1.2x²-3x-5/x+1 2.2x³-13x²-x+3/2x+1

Write the quadratic function in intercept form that has intercepts of -3 and 7 and goes through the point (6, -9)

When the equation 5.31x² +2.92x-1.93=0 is solved, the two values of the unknown x are _____ and ______

Factor 5x² 13x - 6. The factors are

A model rocket is launched vertically from a platform 64 feet above the ground. The height the rocket reaches during the flight is modeled by the equation : s(t) = −16t² + 48t + 64 s is the height of the rocket and t is the time in seconds since the launch. After how many seconds will the rocket hit the ground? 3 2 4 1

Write the quadratic function in standard form that goes through the points (1, 5), (0,3), and (2, 3)

Write the quadratic function in standard form. y = -2(x - 3)² - 3 (fill in the blanks with the correct coefficient. If the coefficient has a "minus sign" in front make the number negative that you enter. Do not enter a "+" for positive numbers.)

Use the quadratic formula to solve for x. 2x²-3x-7=0 (If there is more than one solution, separate them with commas.)

What is the equation for g, which is f(x) = 2x2 + 3x - 1 reflected across the y-axis? g(x) = 2x² + 3x - 1 B g(x) = -2x2 - 3x + 1 g(x) = 2x²-3x - 1 g(x) = -2x²-3x - 1

Solve the equation by the method of your choice. x² +6x=10 The solution set is

Solve the equation using the quadratic formula. 10x²= 2x+3 The solution set is

Solve the equation by the square root property. 3(x+5)² = 54 The solution set is.

Solve the quadratic equation by completing the square. x² - 6x-6=0 To complete the square, what number should be added to both sides of the equation? (Type an integer or a simplified fraction.)

Solve the equation by the square root property. 3(x+6)² = 36 The solution set is (Type an exact answer, using radicals as needed. Use a comma to separate answers as needed. Simplify your answer.)

17. Fill in the blank to complete the next step in solving the quadratic equation using the complete the square method: c² + 10c + 8 = 0 c² + 10c = -8 c²+10c + 25 = -8 +25 (c +. )² = 17 A. 25 B. 10 C. 5 D. -5

When p (x) is divided by (2x - 1), the quotient is (x² + 3x - 1) and the remainder is 10. The function p(x) is: a) p (x) = 2x³ – 5x² − 5x + 1 Ob) p(x) = 2x³ - 5x² + 5x + 11 c) p(x) = 2x³ + 5x² − 5x + 11 d) p(x) = 2x³ + 5x² − 5x + 1

Find the zeros for the given polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis or touches the x-axis and turns around at each zero. f(x)=x³-30x² +225x Determine the zero(s), if they exist. The zero(s) is/are (Type integers or decimals. Use a comma to separate answers as needed.) Determine the multiplicities of the zero(s), if they exist. Select the correct choice below and, if necessary, fill in the answer box(es) within your choice. A. There is one zero. The multiplicity of the zero is (Simplify your answer.) B. There are two zeros. The multiplicity of the smallest zero is (Simplify your answers.) C. There are three zeros. The multiplicity of the smallest zero is (Simplify your answers.) The multiplicity of the largest zero is The multiplicity of the largest zero is The multiplicity of the other zero is choice

Describe the multiplicity and zeros for the following polynomial: (x-9)2(x+8)7(x+10) Roots: . 2 (multiplicity of 9) . 7 (multiplicity of 8) . 1 (multiplicity of 10) Roots: . -9 (multiplicity of 2) .8 (multiplicity of 7) . 10 (multiplicity of 1) Roots: . 9 (multiplicity of 2) . -8 (multiplicity of 7) .-10 (multiplicity of 1) Roots: . 9 (multiplicity of 2) . -8 (multiplicity of 7) .-10 (multiplicity of 0)

When finding the discriminant and it's positive, how many real solutions does it have? What if it's negative? What if it's 0? positive = no real solutions; negative = 1 real solution; 0 = 2 real solutions positive = 2 real solutions; negative 0 real solutions; 0 = 1 real solution positive 2 real solutions; negative = 1 real solution; 0= no real solutions

For the equation below: 6x² + 5x+8=0 Find the discriminant of the quadratic equation. How many real solutions does the quadratic equation have? A Two Real Solutions B One Real Solution c No Real Solutions

Frankie performed the following steps to solve the equation 2m² Step 1: 2m² - 24m +54 = 0 Step 2: 2 (m² - 12m +27) = 0 Step 3: 2 (m - 9) (m - 3) = 0 Step 4: m = -9, -3, 2 Which of the following statements is true? A Frankie solved the equation correctly.. B) Frankie did not perform the correct inverse operation in Step 1. Frankie did not factor out the common factor in Step 2. D) Frankie did not factor the trinomial correctly in Step 3. E) Frankie did not find the correct solutions in Step 4.

Given the height of a rocket is represented by a function given by H (t) = -16t^2 +50t + 50. Where H is in feet and t is in seconds. What is the initial height of the rocket? A 50 feet B 12 feet C 0 feet D -16 feet

Select all the polynomial functions whose graphs have x-intercepts at x = 4,-1/4,-2. a. (x+4)(4x−1)(x−2) b.(x−4)(4x + 1)(x+2) c.(x−4)(4x−1)(x−2) d.(x + 4)(4x + 1)(x+2) e.(2x+4)(4x−1)(x−2) f.(4x-16)(4x + 1)(x+2) Explain or show your reasoning:

To solve a quadratic equation by factoring, please order the following steps: Start 1 2 3 Factor completely. Set each factor equal to zero and solve for x. Get equation set equal to zero.

Consider the polynomials below. P(r) = (x³ + 3x) (2x² - 7x + 4) Q(x) = 215 714 - 1³ + 6x² + 11x - 9 Determine which operation results in the simplified expression below. 8x5-28x4+7x3- 3x² + 45x - 27 A. 5P-Q B. 4P+ Q C. P+3Q D. 2P-Q

Rewrite the following quadratic function in standard (vertex) form. Enter exact values and use improper fractions, if necessary. Provide your answer below: f(x) = -7x² - 4x − 6 f(x) =

Graph the parabola. y = 3x² + 18x+21 Plot five points on the parabola: the verte vertex, and two points to the right of the function button.

A car's velocity is modeled by v(t) = 0.5t² - 14t+ 80 for 0 ≤t≤ 14, where the velocity is in feet per second and time is in seconds. When does the car come to a complete stop? 20 seconds O 14 seconds O8 seconds 4 seconds

Select the correct answer. The first 1,500 students who register cars with the parking facilities office at a state university are guaranteed a parking space. The initial number of students at the office when it opened was 83, and the number of students that visit the office increases every hour by 17%. Which of the following inequalities can be used to determine the number of hours, t, after the parking facilities office opened when the number of students at the office will be at or below 1,500? 83(1.83) 1,500 83(1.017) 1,500 O 83(1.17) 1,500 083 (0.83) > 1,500

Write the quadratic equation in standard form: Answer: 4x14-3x² Submit Answer

An architect has designed two tunnels. Tunnel A is modeled by x2 + y² + 20x - 69 = 0, and tunnel B is modeled by x² - 40x + 16y + 160 = 0, where all measurements are in feet. The architect wants to verify whether a truck that is 8 feet wide and 13.5 feet high can pass through the tunnels. Part A: Write the equation for Tunnel A in standard form and determine the conic section. Show your work. (4 points) Part B: Write the equation for Tunnel B in standard form and determine the conic section. Show your work. (4 points) Part C: Determine the maximum height of each tunnel. Is the truck able to pass through either tunnel without damage? If so, which tunnel(s) and why? Show your work. (7 points)

Solve the quadratic by factoring. 2x²9x13 = -8

What is the discriminant of the quadratic equation 8x2-3x+7=0?

Factor completely. 5x2 -8x+3

What are the roots of the equation 2 - 10x + 26 0 in simplest a + bi form?