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Tool to apply the quadratic formula to any polynomial of degree 2 (ax ^ 2 + bx + c) from the expression of the trinomial or the values of a, b and c.

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Tag(s) : Arithmetics

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### From polynomial expression

 I want to Apply the quadratic formula Solve the equation P=0 Expand the polynomial (expanded form ax^2+bx+c) Extract a, b and c values

### What is the quadratic formula? (Definition)

The quadratic formula is the name given to a mathematical expression allowing to find the solutions of a quadratic equation (presented in the form of a polynomial of degree 2 equal to 0). This is the easiest method and therefore the most often taught.

For any polynomial $P$ of order 2, of variable $x$ denoted $$P(x) = ax^2+bx+c$$ the solutions of $P(x) = 0$ are given by the quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

The formula involves the symbol $\pm$ (plus or minus) which means that $x$ can take 2 values, the first calculated with the sign $+$ (plus), the second with the sign $-$ (minus).

### How to apply the quadratic formula?

From a polynomial of degree 2, of variable $x$:

Expand and reduce the expression of the polynomial if necessary

— Note $a$ the coefficient associated with $x^2$

— Note $b$ the coefficient associated with $x$

— Note $c$ the remaining constant

— Calculate the solutions $$x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a} \\ x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a}$$

Example: $5x^2+3(x+1)-4$ expands to $5x^2+3x-1$, the values of the coefficients are $a = 5, b = 3, c = -1$, the solutions are $x_1 = \frac{-3+\sqrt{29}}{10}$ and $x_2 = \frac{-3-\sqrt{29}}{10}$

If $b^2 - 4ac = 0$ then there is only one solution.

### What is the demonstration of the quadratic formula?

From the initial equation $$ax^2+bx+c = 0$$

— Divide by $a$: $$x^2 + \frac{b}{a} x + \frac{c}{a}=0$$

Subtract $\frac{c}{a}$: $$x^2 + \frac{b}{a} x = -\frac{c}{a}$$

— Apply the square completion, so add $\left( \frac{b}{2a} \right)^2$ to get $$x^2 + \frac{b}{a} x + \left( \frac{b}{2a} \right)^2 = -\frac{c}{a} + \left( \frac{b}{2a} \right)^2$$

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

— Rearrange the fractions on the right over their common denominator $4a^2$: $$\left( x + \frac{b}{2a} \right)^2 = \frac{b^2-4ac}{4a^2}$$

— Calculate the square root: $$x + \frac{b}{2a} = \pm \frac{\sqrt{b^2-4ac}}{2a}$$

— Isolate $x$ to get the final formula: $$x = \frac{ -b \pm \sqrt{b^2-4ac}}{2a}$$

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