Definition
The quadratic formula is used to find the root(s) of a quadratic equation. Roots are also called x-intercepts or zeros. A quadratic function is graphically represented by a parabola with vertex located at the origin, below the x-axis, or above the x-axis. Therefore, a quadratic function may have one, two, or zero roots. When we are asked to solve a quadratic equation, we are really being asked to find the roots.
Quadratic Equation in Standard Form: ax2 + bx + c = 0
- Quadratic Equations can be factored
- Quadratic Formula: x = ( −b ± √(b2 − 4ac) ) / 2a
- When the Discriminant (b2−4ac) is:
- positive, there are 2 real solutions
- zero, there is one real solution
- negative, there are 2 complex solutions
The vertex form of a quadratic equation looks like this: f(x) = a(x – h)^2 + k, where a is not zero and (h, k) is the vertex of the function. This form tells us how high above/below the x-axis the vertex lies (the value of k) and how far left/right of the y-axis the vertex lies (the value of h).
The a in the quadratic vertex form is the same one as in the standard form of a quadratic equation: f(x) = ax2 + bx + c, whereas b = -2ah and c = ah2 + k. [2]
Who
We all have studied quadratic equations. Sorry for my mistake, we did not study but in reality, most of us have just practiced solving quadratic equations in schools, colleges, or even in some basic math courses at the university level. Right?
Jalal Mansoori – “How Can I Use Quadratic Equations In Real Life?“
What
Real-life Applications [1]
- The zeroes of a quadratic are used to find many physical interpretations, such as equilibrium prices in supply-demand applications and objects hitting the ground in physics.
- The vertex is a relative extremum in general, and absolute extremum in the case of a quadratic.
- The axis of symmetry can assist with many graphing applications in art and architecture.
See articles on the use of the quadratic function in the Additional Reading section below.
Why
The quadratic vertex form is useful for several reasons: [2]
- It helps us to find the zeros of the function, with a little bit of algebra (we solve the equation for x with f(x) = 0)
- It is easier to find the vertex of the parabola (we can read the values of h and k from the equation to find the coordinates (h, k) of the vertex)
- It is easier to find points for graphing the parabola (use the vertex (h, k) as a base point, and use the zeros if the function has them, or choose x = 0, f(x) = f(0) = ah2 + k).
See Theoretical Knowledge Vs Practical Application.
How
I don’t show you how to to use the quadratic formula. Many of the References, Additional Reading, websites and YouTube videos will assist you with that. As some professors say: “It is intuitively obvious to even the most casual observer.”
“A Different Way To Solve Quadratic Equations: Loh’S Method”. 2021. Medium. https://mathladyhazel.medium.com/a-different-way-to-solve-quadratic-equations-lohs-method-91fb80c3e645.
I’m going to be talking about a new method of solving quadratic equations, which a maths professor recently announced at Carnegie Mellon University called Po-Shen Loh. You might disagree that this is precisely a new method because it’s more like a new way of looking at the old way of solving quadratic equations.
References
[1] “Find Features of Quadratic Functions”. 2023. Khan Academy Wiki. https://khanacademy.fandom.com/wiki/Find_features_of_quadratic_functions.
[2] Madore, Jonathon. “What is the vertex form of a quadratic equation?”. 2023. Quora. https://qr.ae/pyUInr.
Additional Reading
“101 Uses Of A Quadratic Equation”. 2004. plus.maths.org. https://plus.maths.org/content/os/issue29/features/quadratic/index.
“101 Uses Of A Quadratic Equation: Part II”. 2004. plus.maths.org. https://plus.maths.org/content/101-uses-quadratic-equation-part-ii.
The quadratic equation has played a pivotal part in not only the whole of human civilisation as we know it, but in the possible detection of other alien civilisations and even such vital modern activities as watching satellite television. What else, apart from the nature of divine revelation, could be considered to have had such an impact on life as we know it? Indeed, in a very real sense, quadratic equations can save your life.
“10.3 – Applications Of Quadratic Functions | Hunter College – MATH101”. 2021. courses.lumenlearning.com. https://courses.lumenlearning.com/cuny-hunter-collegealgebra/chapter/applications-of-quadratic-functions/.
Quadratic equations are widely used in science, business, and engineering. Quadratic equations are commonly used in situations where two things are multiplied together and they both depend on the same variable. For example, when working with area, if both dimensions are written in terms of the same variable, you use a quadratic equation. Because the quantity of a product sold often depends on the price, you sometimes use a quadratic equation to represent revenue as a product of the price and the quantity sold. Quadratic equations are also used when gravity is involved, such as the path of a ball or the shape of cables in a suspension bridge.
“Everyday Examples Of Situations To Apply Quadratic Equations”. 2021. Sciencing. https://sciencing.com/everyday-examples-situations-apply-quadratic-equations-10200.html.
Quadratic equations are actually used in everyday life, as when calculating areas, determining a product’s profit or formulating the speed of an object.
⭐ Frankis, Finn. “Why does the quadratic equation work all the time?” 2022. Quora. https://qr.ae/pv9nAI.
It always works because it is simply another form of the equation ax2 + bx + c = 0, solved for x.
“How Can I Use Quadratic Equations In Real Life ?” 2020. Medium. https://medium.com/illumination/how-can-i-use-quadratic-equations-in-real-life-239a70eeb6d0.
We all have studied quadratic equations. Sorry for my mistake, we did not study but in reality, most of us have just practiced solving quadratic equations in schools, colleges, or even in some basic math courses at the university level. Right?
“Real World Examples Of Quadratic Equations”. 2021. mathsisfun.com. https://www.mathsisfun.com/algebra/quadratic-equation-real-world.html.
“Uses Of Quadratic Equations In Daily Life – All Uses Of”. 2019. All Uses Of. https://allusesof.com/math/uses-of-quadratic-equations-in-daily-life/.
Videos
⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.
Mathematical Mysteries 