Solving the Square Equation: ax^2 + bx + c

• Werg22
In summary, The square equation, a(x + {\frac {b} {2a})^{2} + ({c - {\frac {b^2} {4a}) , can be obtained from ax^2 + bx + c by manipulating the terms to create a perfect square. This is done by adding and subtracting a specific term and then factoring it into a squared expression. The significance of this is to have a simplified and easier form of the original equation.
Werg22
I just want to see how can the square equation, $$a(x + {\frac {b} {2a})^{2} + ({c - {\frac {b^2} {4a})$$, can be optained from

$$ax^2 + bx + c$$

Can anyone show me how the equation is manipulated to result into the square form?

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your expression isn't quite right, but the correct one is easy to derive:

$$ax^2+bx+c = a\left( x^2 + \frac{b}{a}x\right) + c = a\left(x^2 + \frac{b}{a}x + \frac{b^2}{4a^2} - \frac{b^2}{4a^2}\right) + c$$

$$= a\left(x^2 + \frac{b}{a}x + \frac{b^2}{4a^2}\right) + c - \frac{b^2}{4a} = a\left(x+\frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right).$$

Thank you!

Whats the significance fo this?

My math teacher often don't explain the logic of anything and having learned the equation just today I was quite disturbed by it and I wanted to "understand" the equation. That's all. I admit I've been quite silly for not figuring it out...

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1. What is the formula for solving a square equation?

The formula for solving a square equation is x = (-b ± √(b^2 - 4ac))/2a.

2. How do you determine the value of a, b, and c in a square equation?

In the equation ax^2 + bx + c, a represents the coefficient of the squared term, b represents the coefficient of the linear term, and c represents the constant term.

3. Can a square equation have two solutions?

Yes, a square equation can have two solutions, also known as roots, when the discriminant (b^2 - 4ac) is greater than zero.

4. Can a square equation have no solutions?

Yes, a square equation can have no solutions when the discriminant (b^2 - 4ac) is less than zero.

5. How do you check if a given value is a solution to a square equation?

To check if a given value is a solution to a square equation, simply plug in the value for x in the equation and see if it satisfies the equation. If it does, then it is a solution.

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