Derive the Almighty formula explaining each steps in details

Question

ScanSolve AI Tutor · Accepted Answer

The "Almighty formula" refers to the quadratic formula, which solves equations of the form $ax^2 + bx + c = 0$ .

Start with the standard quadratic equation.
$ax^2 + bx + c = 0$
Divide the entire equation by $a$ (assuming $a \neq 0$ ) to make the coefficient of $x^2$ equal to 1.
$\frac{ax^2}{a} + \frac{bx}{a} + \frac{c}{a} = \frac{0}{a}$ $x^2 + \frac{b}{a}x + \frac{c}{a} = 0$
Move the constant term $\frac{c}{a}$ to the right side of the equation.
$x^2 + \frac{b}{a}x = -\frac{c}{a}$
Complete the square on the left side.
To do this, add $(\frac{1}{2} \cdot \text{coefficient of } x)^2$ to both sides. The coefficient of $x$ is $\frac{b}{a}$ , so we add $(\frac{1}{2} \cdot \frac{b}{a})^2 = (\frac{b}{2a})^2 = \frac{b^2}{4a^2}$ to both sides.
$x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2$ $x^2 + \frac{b}{a}x + \frac{b^2}{4a^2} = -\frac{c}{a} + \frac{b^2}{4a^2}$
Factor the left side as a perfect square.
$\left(x + \frac{b}{2a}\right)^2 = -\frac{c}{a} + \frac{b^2}{4a^2}$
Combine the terms on the right side by finding a common denominator, which is $4a^2$ .
$\left(x + \frac{b}{2a}\right)^2 = -\frac{c}{a} \cdot \frac{4a}{4a} + \frac{b^2}{4a^2}$ $\left(x + \frac{b}{2a}\right)^2 = -\frac{4ac}{4a^2} + \frac{b^2}{4a^2}$ $\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}$
Take the square root of both sides.
Remember to include both positive and negative roots.
$\sqrt{\left(x + \frac{b}{2a}\right)^2} = \pm\sqrt{\frac{b^2 - 4ac}{4a^2}}$ $x + \frac{b}{2a} = \pm\frac{\sqrt{b^2 - 4ac}}{\sqrt{4a^2}}$ $x + \frac{b}{2a} = \pm\frac{\sqrt{b^2 - 4ac}}{2a}$
Isolate $x$ by subtracting $\frac{b}{2a}$ from both sides.
$x = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a}$ $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Derive the Almighty formula explaining each steps in details

Answer

Step-by-step solution

Start with the standard quadratic equation.

Divide the entire equation by $a$ (assuming $a \neq 0$ ) to make the coefficient of $x^2$ equal to 1.

Move the constant term $\frac{c}{a}$ to the right side of the equation.

Complete the square on the left side.

Factor the left side as a perfect square.

Combine the terms on the right side by finding a common denominator, which is $4a^2$ .

Take the square root of both sides.

Isolate $x$ by subtracting $\frac{b}{2a}$ from both sides.

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Derive the Almighty formula explaining each steps in details

Answer

Step-by-step solution

Start with the standard quadratic equation.

Divide the entire equation by aaa (assuming a≠0a \neq 0a=0) to make the coefficient of x2x^2x2 equal to 1.

Move the constant term ca\frac{c}{a}ac​ to the right side of the equation.

Complete the square on the left side.

Factor the left side as a perfect square.

Combine the terms on the right side by finding a common denominator, which is 4a24a^24a2.

Take the square root of both sides.

Isolate xxx by subtracting b2a\frac{b}{2a}2ab​ from both sides.

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More Mathematics Questions

Divide the entire equation by $a$ (assuming $a \neq 0$ ) to make the coefficient of $x^2$ equal to 1.

Move the constant term $\frac{c}{a}$ to the right side of the equation.

Combine the terms on the right side by finding a common denominator, which is $4a^2$ .

Isolate $x$ by subtracting $\frac{b}{2a}$ from both sides.