### Video Transcript

The function drawn below represents a quadratic π of π₯ equals negative π₯ squared plus three π₯ plus π minus two which intersects the π₯-axis at the points π΄ and π΅. Find the value of π given ππ΅ is four times ππ΄.

We have a quadratic function with a negative leading coefficient. That gives us the inverted parabola shape shown. Weβre also told that ππ΅ is equal to four times ππ΄. So letβs define the distance ππ΄ to be π units. And therefore, the distance ππ΅ is four π units. Now, what that actually means is that the coordinate π΄ can be represented as the point negative π zero, whereas the coordinate π΅ must be four π zero. So weβre essentially going to work backwards to find the value of π and ultimately π. The roots of our equation are negative π and four π. So this means when we factor the expression for π of π₯ and set it equal to zero, we should have something along the lines of negative π₯ minus π times π₯ minus four π.

Now, the reason Iβve chosen negative π₯ and π₯ is because I know the coefficient of π₯ squared to be negative one. And if I was going to go back and solve this equation, I would say that either negative π₯ minus π equals zero or π₯ minus four π equals zero. And when I solve each of these four π₯, I get π₯ equals negative π, thatβs our first route, and π₯ equals four π; thatβs our second route as required. So what we can say then is that our function, when factored, must look like this. Itβs negative π₯ minus π times π₯ minus four π. Letβs distribute these parentheses. We multiply the first term in each expression, negative π₯ times π₯ is negative π₯ squared. And we then multiply the outer terms, negative π₯ times negative four π is four π₯π.

Multiplying the inner terms gives us negative π₯π and multiplying the last terms gives us four π squared. We collect like terms, and we see that this simplifies to negative π₯ squared plus three π₯π plus four π squared. But of course, the question tells us that our function is equal to negative π₯ squared plus three π₯ plus π minus two. So letβs equate coefficients and see if we can work out the value of π and, therefore, the value of π. Weβll begin by equating the coefficients of π₯ or π₯ to the power of one. On the left-hand side, the coefficient of π₯ is three π. And on the right-hand side, itβs three. If three π is equal to three than π must be equal to one. And of course, we could solve this equation more formally by dividing both sides by three.

Weβll now equate the constants. Thatβs the coefficient of π₯ to the power of zero. On the left-hand side, we have four π squared. And on the right-hand side, our constant term is π minus two. Now, of course, we have just worked out that π is equal to one. So we have four times one squared equals π minus two or simply four equals π minus two. We solve this equation, for π, by adding two to both sides. And we find π to be equal to six. So given the function negative π₯ squared plus three π₯ plus π minus two such that ππ΅ equals four ππ΄, we find π to be equal to six.