algebra 1a - unit 6: quadratic equations, part 2

lesson 27

the quadratic formula

which statement is true?

a system of equations consists of two or more equations that must be true at the same time.

lesson 26

graphing quadratics in vertex form

what is the solution to the problem?

the area will be 110 square centimeters when the base is about 6.4 cm.

what is the equation of the line of symmetry for the parabola represented by the equation y = −2(x − 3)^2 + 4

x = 3

how many real solutions does the equation below have? 3x^2 − x + 5 = 0

0

the height of a triangle is 1.95 centimeters less than 2.5 times the corresponding base. the area of the triangle is 112.8 square centimeters. the quadratic equation that correctly models this situation is 2.5x^2 − 1.95x = 225.6 or 2.5x^2 − 1.95x − 225.6 = 0, where x represents the base of the triangle. how long is the base of this triangle?

9.90

nathan launches a water balloon vertically from a platform on his roof. the height of the balloon (in feet) is represented by the equation h = −16t^2 + 28.7t + 32.4, where t is the time (in seconds) after he launches the water balloon. what is the maximum height of the balloon?

45.27​ feet

consider the quadratic equation 8 − 3x^2 − 16x = 4 what are the values of a, b, and c when you write this equation in standard form?

a = -3, b = -16, and c = 4

consider the quadratic equation 14x^2 − 17x − 6 = 0 question a: what is the value of the discriminant? question b: how would you describe the solutions?

a: 625 b: two real solutions

graham, an alien from another planet, shoots an arrow straight into the air 2.4 meters from the ground at 36 meters per second. according to the laws of physics on his planet, the gravity is around 2.78 meters per second squared. the height of the arrow (in meters) is represented by the equation p = −2.78t^2 + 36t + 2.4, where t is the time (in seconds) after the arrow is launched. how long does it take the arrow to hit the ground?

about 13.02 seconds

on which of the previous lines did eryka make her first mistake, if she made any?

she made no mistakes.

lesson 28

solving quadratics with technology

lesson 29

systems involving quadratic equations

which statement is true regarding the solution to the problem, as found by substitution?

the ball will be 80 meters above ground level twice after it is kicked: at about 0.58 seconds and about 3.5 seconds.

which statement is true?

the solution to the system is (0, 6).

which statement is true?

the solutions to the system are (−2.6, 3.7) and (3.1, 6.5).

use the substitution method to solve the system of equations {y = −x^2 − 1 ,x + 3y = 6} which statement is true?

there is no solution to the system.

which quadratic equation in factored form could represent this parabola? https://cdstools.flipswitch.com/asset/media/960871

y = 4.9(x − 21)(x − 18)

a chopstick model of a catapult launches a marshmallow in a classroom. the path of the marshmallow can be modeled by the quadratic equation y = −0.05x^2 + x + 1.89, where y represents the height of the marshmallow in feet and x represents the horizontal distance from the point it is launched in feet. at what distance is the marshmallow when it is at a height of 0?

​21.74 feet

consider the quadratic equation x^2 + x + 4 = 0. question a: what is the value of the discriminant? question b: how would you describe the solutions?

a: -15 b: no real solution

renate launched an object vertically from a point that is 58.9 meters above ground level with an initial velocity of 21.6 meters per second. this situation can be represented by the equation h = −4.9t^2 + 21.6t + 58.9, where h is the height of the object in meters and t is the time in seconds after the object is launched. what is the maximum height of the object?

82.7 meters

an object launched vertically from a point that is 37.5 feet above ground level with an initial velocity of 48.6 feet per second can be represented by the equation h = −16t^2 + 48.6t +37.5, where h is the height of the object and t is the time after the object is launched. how long does it take the object to hit the ground?

about 3.68 seconds

what mistake, if any, did agathe make?

agathe didn't make a mistake.

which part of the formula represents the discriminant?

b^2 − 4ac

mark dropped an object from a bridge 400 feet above ground level with an initial velocity of 0. he knows that the gravitational pull of the earth is about 16 feet per second squared. he wants to find how many seconds, t, it will take the object to hit the ground. match each expression with the correct equation that models this situation or solution.

correct equation : -16t^2 + 0t + 400 = 0 solution : 5 seconds

the height of a triangle is 1 less than 4 times the corresponding base. if the area of the triangle is 69 square meters, what is the base of the triangle? match each expression with the correct equation in standard form that models this situation, where b represents the base of the triangle, or solution.

correct equation : 4b^2 − b − 138 = 0 solution : 6 meters

which parabola represents the equation y = −(x − 3)^2 + 9?

https://cdstools.flipswitch.com/asset/media/959768

which parabola represents the equation y = 2(x + 3)^2 − 6?

https://cdstools.flipswitch.com/asset/media/959787

match each parabola with the coordinates of its vertex.

https://cdstools.flipswitch.com/asset/media/959834 : (-1, -4) https://cdstools.flipswitch.com/asset/media/959836 : (1, 4) https://cdstools.flipswitch.com/asset/media/959837 : (3, 0) https://cdstools.flipswitch.com/asset/media/959838 : (4, -1)

match each parabola with the coordinates of its zeros.

https://cdstools.flipswitch.com/asset/media/959834 : (-5, 0) and (3, 0) https://cdstools.flipswitch.com/asset/media/959836 : (-1, 0) and (3, 0) https://cdstools.flipswitch.com/asset/media/959837 : (3, 0) https://cdstools.flipswitch.com/asset/media/959838 : no zeros

match each parabola with the equation it represents.

https://cdstools.flipswitch.com/asset/media/959834 : y = 1/4(x + 1)^2 - 4 https://cdstools.flipswitch.com/asset/media/959836 : y = -(x - 1)^2 + 4 https://cdstools.flipswitch.com/asset/media/959837 : y = 2(x - 3)^2 https://cdstools.flipswitch.com/asset/media/959838 : y = -1/2(x - 4)^2 - 1

tanja kicks a soccer ball with an initial velocity from ground level. the ball reaches its maximum height of 16 feet in 1 second. it hits the ground in 2 seconds. which graph correctly models the height of the ball while airborne?

https://cdstools.flipswitch.com/asset/media/960118

which graph correctly represents the quadratic equation y = 0.0012x^2 − 0.50x + 25.98?

https://cdstools.flipswitch.com/asset/media/960865

which graph correctly represents the quadratic equation y = 1.25(x − 4.8)^2 + 2.17?

https://cdstools.flipswitch.com/asset/media/960873

which graph correctly represents the quadratic equation y = −0.0005x^2 + 1.2345x + −2.2?

https://cdstools.flipswitch.com/asset/media/965928

a soccer ball is kicked with an initial velocity from ground level. the height of the ball can be modeled by the following parabola. match each point with the correct description.

point a : this point represents a zero of the model. it means that the ball is kicked directly from the ground. point b : this point represents the vertex of the model. it means that the ball reaches its maximum height at 7.56 feet in about 0.69 seconds. point c : this point represents a zero of the model. it means that the ball hits the ground in about 1.38 seconds.

a scientist dropped an object from a height of 320 feet. the height of the object is modeled by the equation h = 320 − 16t^2 after how much time will the object hit the ground?

√20 seconds

oscar throws an object to his friend from a point that is above ground level, with an initial velocity. unfortunately, his friend cannot catch the object, and it hits the ground. the quadratic equation h = −16t^2 + 35t + 4 models the height of the object, where h represents its height and t represents the number of seconds the object is in the air. the graph of this equation follows. which statements are true regarding the parabola and this situation?

the y-intercept shows that the object is thrown from 4 feet above ground level. the y-coordinate of the vertex shows that the maximum height the object reaches while airborne is 23.14 feet. the zero, or the xx-intercept, shows that it takes the object about 2.3 seconds to hit the ground after it is thrown.

the length of a rectangle is 2 less than twice its width. the area of the rectangle is 144 square centimeters. which quadratic equation in standard form correctly models this situation, where w represents the width of the rectangle?

w^2 − w − 72 = 0

use the quadratic formula to solve for x: 3x^2 + 14x = −11

x = -11/3 and x = -1

what is the equation of the line of symmetry for the parabola represented by the equation y = −x^2 + 2x + 6?

x = 1

what are the solutions to the equation x^2 − 28x + 160 = 144?

x = 14 + 6√5; x = 14 − 6√5

consider the equation 3x^2 − 2x + 7 = 0 in standard form. which equation shows the coefficients a, b, and c correctly substituted into the quadratic formula?

x = 2 ± √(−2)^2 − 4(3)(7)

what is the equation of the line of symmetry for the parabola represented by the equation y = −2x^2 + 20x − 44?

x = 5

what is the equation of the line of symmetry for the parabola represented by the equation y = −2(x − 7)^2 + 11?

x = 7

use the quadratic formula to solve the quadratic equation 2x^2 + 4x − 1 = 0. what are the solutions to the equation?

x = −2 − √6/2 and x = −2 + √6/2

what is the vertex form of the equation that represents this parabola? https://cdstools.flipswitch.com/asset/media/960055

y = (x + 1)^2 − 9

match each equation with the parabola it represents.

y = −(x + 4)^2 + 5 : https://cdstools.flipswitch.com/asset/media/960018 y = (x + 5)^2 - 4 : https://cdstools.flipswitch.com/asset/media/960020 y = (x − 4)^2 − 5 : https://cdstools.flipswitch.com/asset/media/960021

what is the vertex form of the equation that represents this parabola? https://cdstools.flipswitch.com/asset/media/960045

y = −2(x − 4)^2 + 8

a remote-controlled airplane started to fly from a point above ground level. it flew for a while in the air and then stopped. the following parabola represents its height during the time it was flying. match each part of the parabola with the correct description.

y-coordinate of point a : it is the y-coordinate of the y-intercept. it means that the airplane started to fly at a height of 26 feet above ground level. y-coordinate of point b : it is the y-coordinate of the vertex. it means that the minimum height that the airplane had was 10 feet. x-coordinate of vertex b : it is the x-coordinate of the vertex. it means that the airplane reached its minimum height in about 1.43 seconds. y-coordinate of point c : it is the y-coordinate of one of the end points of the parabola. it means that the airplane was at a height of about 29.36 feet when it stopped flying. x-coordinate of point c : it is the x-coordinate of one of the end points of the parabola. it means that the airplane flew for about 3 seconds before it stopped flying.

which statements are true regarding the quadratic formula?

−b/2a is the x-coordinate of the vertex for the parabola. x = −b/2a is the equation of the line of symmetry of the parabola. ... is the horizontal distance between the line of symmetry and one of the zeros of the parabola.