# Solving Word Problems Using Quadratic Equations

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Again, we can use the vertex to find the maximum or the minimum values, and roots to find solutions to quadratics.

Note that we did a Quadratic Inequality Real World Example here.

Thus, it is concluded that the differences in the structural properties of the symbolic equations and word problem representations affected student performance in formulating and solving quadratic equations with one unknown.

In this lesson I will teach you about quadratic equation word problems.

Note also that we will discuss Optimization Problems using Calculus in the Optimization section here.

where $$t$$ is the time in seconds, and $$h$$ is the height of the ball.

1/t 1/(t-1min) = 76/360 This leads to 19*t^2 - 180*t 71 = 0 (steps suppressed) The answer is ~9.06 min, the other is less than half a min.

Based on this can you find a discrepancy between your approach and mine?

If the first one walks $v$ km/hour, he takes $\frac v$ minutes to walk

where $$t$$ is the time in seconds, and $$h$$ is the height of the ball.

1/t 1/(t-1min) = 76/360 This leads to 19*t^2 - 180*t 71 = 0 (steps suppressed) The answer is ~9.06 min, the other is less than half a min.

Based on this can you find a discrepancy between your approach and mine?

If the first one walks $v$ km/hour, he takes $\frac v$ minutes to walk $1$ km.

The second fact is that it takes the second pedestrian one more minute than the first to cover 1 km, so you have $$\frac1 1=\frac1.$$ Solve the two equations for $v_1$ and $v_2$ and then compute $1/v_1$ and $1/v_2=1/v_1 1$, or substitute $t_1=1/v_1$ and $t_2=1/v_2$ into the two equations and solve for the times directly.

||

where $$t$$ is the time in seconds, and $$h$$ is the height of the ball.1/t 1/(t-1min) = 76/360 This leads to 19*t^2 - 180*t 71 = 0 (steps suppressed) The answer is ~9.06 min, the other is less than half a min.Based on this can you find a discrepancy between your approach and mine? If the first one walks $v$ km/hour, he takes $\frac v$ minutes to walk $1$ km.The second fact is that it takes the second pedestrian one more minute than the first to cover 1 km, so you have $$\frac1 1=\frac1.$$ Solve the two equations for $v_1$ and $v_2$ and then compute $1/v_1$ and $1/v_2=1/v_1 1$, or substitute $t_1=1/v_1$ and $t_2=1/v_2$ into the two equations and solve for the times directly.There will be two solutions to this system of equations, but one of them doesn’t make physical sense for this problem, so that one will be rejected. So I will reproduce what you have with slightly different notation.I am answering because I do not have comment ability yet. Here is the problem, are you trying to get time or rate? d1 d2 = 76km d1 = v1*T d2 = v2*T T = 6h = 360min v1 = 1km/t v2 = 1km/(t-1min) Putting together...What is the maximum height the ball reaches, and how far (horizontally) from Audrey does is the ball at its maximum height?How far does the ball travel before it hits the ground?Solution: Note that in this problem, the $$x$$-axis is measuring the horizontal distance of the path of the ball, not the time, so when we draw the parabola, it’s a true indication of the trajectory or path of the ball.Note also that the equation given is in vertex form (if we add Since the quadratic is already in vertex form ($$y=a k$$, where $$(h,k)$$ is the vertex), we can see that the vertex from $$0=-0.018 8$$ is $$(20,8)$$.

$km. The second fact is that it takes the second pedestrian one more minute than the first to cover 1 km, so you have $$\frac1 1=\frac1.$$ Solve the two equations for$v_1$and$v_2$and then compute where $$t$$ is the time in seconds, and $$h$$ is the height of the ball. 1/t 1/(t-1min) = 76/360 This leads to 19*t^2 - 180*t 71 = 0 (steps suppressed) The answer is ~9.06 min, the other is less than half a min. Based on this can you find a discrepancy between your approach and mine? If the first one walks$v$km/hour, he takes$\frac v$minutes to walk$1$km. The second fact is that it takes the second pedestrian one more minute than the first to cover 1 km, so you have $$\frac1 1=\frac1.$$ Solve the two equations for$v_1$and$v_2$and then compute$1/v_1$and$1/v_2=1/v_1 1$, or substitute$t_1=1/v_1$and$t_2=1/v_2$into the two equations and solve for the times directly. || where $$t$$ is the time in seconds, and $$h$$ is the height of the ball.1/t 1/(t-1min) = 76/360 This leads to 19*t^2 - 180*t 71 = 0 (steps suppressed) The answer is ~9.06 min, the other is less than half a min.Based on this can you find a discrepancy between your approach and mine? If the first one walks$v$km/hour, he takes$\frac v$minutes to walk$1$km.The second fact is that it takes the second pedestrian one more minute than the first to cover 1 km, so you have $$\frac1 1=\frac1.$$ Solve the two equations for$v_1$and$v_2$and then compute$1/v_1$and$1/v_2=1/v_1 1$, or substitute$t_1=1/v_1$and$t_2=1/v_2$into the two equations and solve for the times directly.There will be two solutions to this system of equations, but one of them doesn’t make physical sense for this problem, so that one will be rejected. So I will reproduce what you have with slightly different notation.I am answering because I do not have comment ability yet. Here is the problem, are you trying to get time or rate? d1 d2 = 76km d1 = v1*T d2 = v2*T T = 6h = 360min v1 = 1km/t v2 = 1km/(t-1min) Putting together...What is the maximum height the ball reaches, and how far (horizontally) from Audrey does is the ball at its maximum height?How far does the ball travel before it hits the ground?Solution: Note that in this problem, the $$x$$-axis is measuring the horizontal distance of the path of the ball, not the time, so when we draw the parabola, it’s a true indication of the trajectory or path of the ball.Note also that the equation given is in vertex form (if we add Since the quadratic is already in vertex form ($$y=a k$$, where $$(h,k)$$ is the vertex), we can see that the vertex from $$0=-0.018 8$$ is $$(20,8)$$. /v_1$ and

where $$t$$ is the time in seconds, and $$h$$ is the height of the ball.

1/t 1/(t-1min) = 76/360 This leads to 19*t^2 - 180*t 71 = 0 (steps suppressed) The answer is ~9.06 min, the other is less than half a min.

Based on this can you find a discrepancy between your approach and mine?

If the first one walks $v$ km/hour, he takes $\frac v$ minutes to walk $1$ km.

The second fact is that it takes the second pedestrian one more minute than the first to cover 1 km, so you have $$\frac1 1=\frac1.$$ Solve the two equations for $v_1$ and $v_2$ and then compute $1/v_1$ and $1/v_2=1/v_1 1$, or substitute $t_1=1/v_1$ and $t_2=1/v_2$ into the two equations and solve for the times directly.

||

where $$t$$ is the time in seconds, and $$h$$ is the height of the ball.1/t 1/(t-1min) = 76/360 This leads to 19*t^2 - 180*t 71 = 0 (steps suppressed) The answer is ~9.06 min, the other is less than half a min.Based on this can you find a discrepancy between your approach and mine? If the first one walks $v$ km/hour, he takes $\frac v$ minutes to walk $1$ km.The second fact is that it takes the second pedestrian one more minute than the first to cover 1 km, so you have $$\frac1 1=\frac1.$$ Solve the two equations for $v_1$ and $v_2$ and then compute $1/v_1$ and $1/v_2=1/v_1 1$, or substitute $t_1=1/v_1$ and $t_2=1/v_2$ into the two equations and solve for the times directly.There will be two solutions to this system of equations, but one of them doesn’t make physical sense for this problem, so that one will be rejected. So I will reproduce what you have with slightly different notation.I am answering because I do not have comment ability yet. Here is the problem, are you trying to get time or rate? d1 d2 = 76km d1 = v1*T d2 = v2*T T = 6h = 360min v1 = 1km/t v2 = 1km/(t-1min) Putting together...What is the maximum height the ball reaches, and how far (horizontally) from Audrey does is the ball at its maximum height?How far does the ball travel before it hits the ground?Solution: Note that in this problem, the $$x$$-axis is measuring the horizontal distance of the path of the ball, not the time, so when we draw the parabola, it’s a true indication of the trajectory or path of the ball.Note also that the equation given is in vertex form (if we add Since the quadratic is already in vertex form ($$y=a k$$, where $$(h,k)$$ is the vertex), we can see that the vertex from $$0=-0.018 8$$ is $$(20,8)$$.

/v_2=1/v_1 1$, or substitute$t_1=1/v_1$and$t_2=1/v_2\$ into the two equations and solve for the times directly.

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