Grades 11-12 Video Solutions 2021-video 2021 11-12/10

video 2021 11-12/10

Video Transcription

Problem number 10 states, the parabola in the figure has an equation of the form ax squared plus bx plus c for some distinct real numbers a, b, and c.  Which of the following equations could be an equation of the line in the figure?  So here we have the diagram of the parabola with equation ax squared plus bx plus c  intersecting with line dx plus e, where we need to figure out what d and e are.  To start out, let's figure out what happens at x equals zero. This would mean that x equals zero in the two equations. Plugging that in, this is what we get.  Luckily, anything times zero is zero, so a lot of it cancels out, and we end up getting that y equals c and y equals e. We can see that at x equals zero,  the two equations meet, so their y values are the same, so we can set them equal to each other.  So we've figured out what e is. It is c.  So let's take a look at our answers.  We can disregard all the ones that don't have plus c, which leaves us with only answers a and d, which have slopes of b or a.  So let's take a look at the second intersection point. This one is going to occur at y equals zero.  So let's solve for the x value.  We get minus c over d equals x. We know d can either be a or b, so we get either minus c over b or minus c over a.  Let's plug that back into the equation for the quadratic and see what we get.  First, let's assume that x equals minus c over b and simplify the equation. Now let's cancel out the minus c and the plus c,  and in the end, we get that ac squared over b squared is equal to zero.  Let's take a look at the sketch of the parabola again. It opens upwards, which means that a must be greater than zero,  and it crosses the y-axis above the x-axis, which means that c also must be greater than zero.  This means that it is impossible for a times c squared to be equal to zero. This means the initial assumption that x equals minus c over b is wrong,  so x must equal minus c over a, and let's see if this works.  Simplifying this is what we get. Multiplying both sides by a, we get this.  Remembering that c is greater than zero, which means it's not zero, which means we can divide by it, so in the end, we get b equals c plus a.  Plugging that back into the parabola equation, we get this. Distributing, we get this, and we notice that we can factor an ax from the left and a c from the right, and we get this.  Factoring those out, we get this, and now we can solve for x,  and we get our value of minus c over a, which means that it is possible for this to be an intercept.  Now let's just solve for the slope again, and we get y equals ax plus c, which is our final answer.  So the question asked us, which of the following equations could be an equation of the line in the figure?  The answer is y equals ax plus c, letter d.

Video Summary

The transcript describes solving for the equation of a line that intersects a parabola with the equation \( ax^2 + bx + c \). To find the intersecting line's equation, first examine \( x = 0 \), resulting in the line's constant term matching the parabola's \( c \). This limits possible equations to those ending in \( + c \). Evaluating further at \( y = 0 \) identifies two potential slopes: \( b \) and \( a \). The calculation eliminates \( b \) due to constraints on \( a \) and \( c \), establishing \( y = ax + c \) as the correct equation for the intersecting line.