Algebra: Understanding Quadratic Equations
A quadratic function is defined by the general polynomial ax2 + bx + c. if we plot this function on a graph we should get a parabola. The shape of the parabola solely depends on the sign before the ‘a’ value. If the sign is positive (+) then you will have graph/parabola that is shaped a cup, if the sign is negative (-), then you will have a graph/parabola shaped like a hat. Quadratic equations/functions are very much applicable to the real world context, especially when it comes on to concepts such as velocity, acceleration, average speed and speed. However, for the purpose of the SAT exam, you will see questions that will require you to derive a quadratic function from a scenario, as well as using either the a × c method or the quadratic formula.
Before we get to solving quadratic equations, there are some background information that should be considered. In the general formula given above, ‘a’ is the always the coefficient of x2 and ‘b’ is the coefficient of the variable x, and ‘c’ is the always the constant value. Thus the function f¸ such that f(x) = x2 + 3x + 2, the coefficient of x2 is 1, the coefficient of x is 3, while the constant value is 2. Knowing this is critical when factorizing a quadratic function in order to solve for x, using the both the a × c method and quadratic formula. One should also note that the quadratic formula can be used to solve any quadratic functions, however, the use method is preferred if the function/equation can be factorized.
What is the value of x in the following equation: x2 + 7x + 12 = 0?
In order to solve this problem, you must first check if the a × c method can be applied, that is, if it can be factorized. Therefore we look at the general formula and identify the a, b, and c value. From the question, we can see that a = 1, b = 7 and c = 12. We know if an equation can be factorized if you multiply the ‘a’ value times the c value. If we call that value m, then a × c = m. when then check if when can find two factors of m, if we add them we should get the ‘b’ value. So let us call us, the factors e and d. thus,
Two factors of m: e and d, as such, e + d = b. Another thing we must be in mind is that b × d = m, this means you must also be able to multiply the factors to get back the results of a × c. The essence of doing this is to rewrite the equation in a different form so that you can factorize. Now let apply all that concept to the question. a × c = 1 × 12 = 12. Now list all the factors of 12: 1, 2, 3, 4, 6, 12. We need to identify two factors that’s satisfies e + d = 7 and b × d = 12. From the factors we could see that if we add 3 and 4 we get 7 which is the b value and if we multiply them we get 12 which is result of a × c. once we can identify the factors, we rewrite the equation substituting the factors for the b value, then factorize. Thus:
x2 + 4x + 3x + 12 = 0
x(x + 4) +3(x + 4) = 0
(x + 4)(x + 3) = 0
x + 4 = 0, x + 3 = 0
x = -4 and x = -3
Therefore the values for x are -4 and -3. These values are the points on the graph of which the parabola cuts the x – axis, then it is the line y = 0.
The quadratic formulae could also be used:
Therefore, using the formula and the a × c method should produce the same answer. One should also note that there are some quadratic functions that cannot be factorized and you would have to use the formula. One should also note that it is not enough to just know how to work a question without understanding the concept and as such, there is great emphasis on conceptual understanding.
A ball is thrown straight up, from 3 m above the ground, with a velocity of 14 m/s. Which of the following equations can be used to get the height at which the ball hits the ground?
(a) h = 3 + 14t + 5t2
(b) h = 3 + 14t – 5t2
(c) h = 3 – 14t – 5t2
(d) h = 5t2 – 14t – 3
When approaching this question, it best of we consider what we know, that is, what information is provided in the question and what concept is it connected to? We know that the height at which the ball is thrown from above the ground which is 3 m. We know that the ball travelled at 14 m/s in the air, and we also know that the ball came down to hit the ground. The next question we ask ourselves is, what do we want to know? We want to know the height of which the ball hits the ground and let us call that h. In order to get h, we must add the distance at which the ball travelled from (3 m), with the distance it travelled into the air before it started going downwards (14 m/s), with the distance from which it started to fall to when it hits the ground (unknown). We will also consider seconds as time (t). Therefore it is 3 + 14t.
To get the distance at which is travelled from in the air to the ground, we have to consider that gravity is acting upon the ball. Therefore, the distance at which it travelled it downwards is defined by s = ut – (1/2at2), where ‘a’ = 10 m/s2 which is the gravitation pull on the earth, some books will say 9.8 m/s2. ‘u’ will be the initial velocity, but because it will come to a stop instantaneously in the air, then fall to the earth, it is zero. ‘s’ is the distance, which we don’t know. We have meters per second squared because it is acceleration, and it is negative because it is coming down, which suggests that the ball is travelling backwards since it was thrown up initially. Therefore we get s = 0(t) – (1/2 × 10 m/s2) = -5 m/s2. Therefore is 3 + 14t + (- 5t2) = 3 + 14t – 5t2, therefore the answer is (b). The SAT exam requires you to connect different concepts to solve the questions. So make sure you practice as much as you can with different questions.
Using the same question, and since we know the ‘a’, ‘b’ and ‘c’ value we can just substitute in the equation: