Proof of the Quadratic Formula – korean math
Hello, I am Teacher Jojo, teaching mathematics in Korea.
Today, let’s prove the quadratic formula, which is used to find the roots of quadratic equations.
There are two methods for solving quadratic equations.
The first one is using factorization, and the second one is using the quadratic formula.
If you understand these two methods, you can find the solutions to all quadratic equations.
In this post, let’s prove the quadratic formula.
[Proof of the Quadratic Formula]
When proving, it can be challenging for students if we proceed directly with symbols.
Let’s start by solving a typical quadratic equation first.
{\color{blue} Step\; 1)} Let’s find the roots of this quadratic equation.
x^2 = 5
Solution)
x = \pm \sqrt{5}
{\color{blue} Step\; 2)} Let’s find the roots of this quadratic equation.
(x + 2)^2 = 3
Solution)
x + 2 = \pm \sqrt{3}
x = -2 \pm \sqrt{3}
{\color{blue} Step\; 3)} Let’s find the roots of this quadratic equation.
2x^2 + 4x – 7 = 0
Solution)
{\color{green}2(x^2 + 2x)} – 7 = 0
{\color{green}2(x + 1)^2 – 2} – 7 = 0
2(x + 1)^2 – 9 = 0
2(x + 1)^2 = 9
(x + 1)^2 = \cfrac{9}{2}
x + 1 = \pm \sqrt{ \cfrac{9}{2} }
x +1 = \pm \cfrac {3 \sqrt{2}}{2}
x = -1 \pm \cfrac {3 \sqrt{2}}{2} = \cfrac {-2 \pm 3 \sqrt{2}}{2}
If you have solved the problem up to step 3 with understanding, you can prove the quadratic formula.
It might seem challenging at first, but take it slowly and follow along.
[Proof of the Quadratic Formula]
Let’s find the roots of this quadratic equation.
ax^2 + bx + c = 0
{\color{green} a(x^2 + \cfrac{b}{a} x)} + c = 0
{\color{green} a(x + \cfrac{b}{2a})^2 \; – \; \cfrac{b^2}{4a}} + c = 0
a(x + \cfrac{b}{2a})^2 = \cfrac{b^2}{4a} \; – \; c
a(x + \cfrac{b}{2a})^2 = \cfrac{b^2-4ac}{4a}
{\color{green} \cfrac{1}{a}} \times a(x + \cfrac{b}{2a})^2 = \cfrac{b^2-4ac}{4a} \times {\color{green} \cfrac{1}{a}}
(x + \cfrac{b}{2a})^2 = \cfrac {b^2-4ac}{4a^2}
x + \cfrac{b}{2a} = \pm \sqrt{ \cfrac{b^2-4ac}{4a^2}}
x + \cfrac{b}{2a} = \pm \cfrac{ \sqrt{b^2-4ac}}{2a}
x = – \cfrac{b}{2a} \pm \cfrac{ \sqrt{b^2-4ac}}{2a}
{\color{red} x = \cfrac { -b \pm \sqrt{b^2-4ac}}{2a}}
The proof of the quadratic formula is complete.
How did it go?
Were you able to follow along?
Take your time, and you can do it too.
On my website, you’ll find a variety of math problems.
Take a look.
Quadratic Inequality Question – Korean Math #22
Also, my YouTube channel is
https://www.youtube.com/channel/UCnJ-GLzfJdWjs04eQoxfY7g
[Korean ver]
안녕하세요. 저는 한국에서 수학을 가르치고있는 조조쌤입니다.
오늘은 이차방정식의 근을 구할때, 근을 쉽게 찾는 공식인 근의공식을 증명해보도록하겠습니다.
완전제곱식과 제곱근의 성질을 알고있다면 이 증명을 이해하는데 도움이 될것입니다.
그럼 시작해볼까요
[증명]
바로 증명을 시작하면 어려울수있습니다.
단계별로 실제 예를 통해서 근을 구해보고, 그다음 문자로 표현을하며 근의 공식을 유도해보도록합시다.
{\color{blue} Step\; 1)}
x^2 = 5 의 근을 구해봅시다.
Solution)
x = \pm \sqrt{5} 입니다. 쉽죠?
{\color{blue} Step\; 2)}
(x + 2)^2 = 3 의 근을 구해봅시다.
Solution)
x + 2 = \pm \sqrt{3}
x = -2 \pm \sqrt{3}입니다. 할만하죠?
{\color{blue} Step\; 3)}
2x^2 + 4x – 7 = 0
Solution)
{\color{green}2(x^2 + 2x)} – 7 = 0
{\color{green}2(x + 1)^2 – 2} – 7 = 0
2(x + 1)^2 – 9 = 0
2(x + 1)^2 = 9
(x + 1)^2 = \cfrac{9}{2}
x + 1 = \pm \sqrt{ \cfrac{9}{2} }
x +1 = \pm \cfrac {3 \sqrt{2}}{2}
x = -1 \pm \cfrac {3 \sqrt{2}}{2} = \cfrac {-2 \pm 3 \sqrt{2}}{2}
조금 복잡하긴 하지만 할만하죠?
3단계까지 잘 따라왔다면 이제 숫자가아닌 문자가 포함되어있을때도 보도록하죠.
이제 시작합니다.
[Proof of the Quadratic Formula]
ax^2 + bx + c = 0
{\color{green} a(x^2 + \cfrac{b}{a} x)} + c = 0
{\color{green} a(x + \cfrac{b}{2a})^2 \; – \; \cfrac{b^2}{4a}} + c = 0
a(x + \cfrac{b}{2a})^2 = \cfrac{b^2}{4a} \; – \; c
a(x + \cfrac{b}{2a})^2 = \cfrac{b^2-4ac}{4a}
{\color{green} \cfrac{1}{a}} \times a(x + \cfrac{b}{2a})^2 = \cfrac{b^2-4ac}{4a} \times {\color{green} \cfrac{1}{a}}
(x + \cfrac{b}{2a})^2 = \cfrac {b^2-4ac}{4a^2}
x + \cfrac{b}{2a} = \pm \sqrt{ \cfrac{b^2-4ac}{4a^2}}
x + \cfrac{b}{2a} = \pm \cfrac{ \sqrt{b^2-4ac}}{2a}
x = – \cfrac{b}{2a} \pm \cfrac{ \sqrt{b^2-4ac}}{2a}
{\color{red} x = \cfrac { -b \pm \sqrt{b^2-4ac}}{2a}}
이렇게 증명이 마무리가되었습니다.
완전제곱식을 이용하고 제곱근을 구하는 방법으로 유도해 나가는 것입니다.
만약 이해가 안된다면
동영상으로도 보실수있습니다.
왜냐면 한국어로 제가 찍어놓은 동영상이 있기 때문이죠.
링크 걸어둡니다. 참고하실분은 보세요.
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