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### What is a Quadratic Equation?

A Quadratic Equation is any equation having the form
$$ax^2 + bx + c = 0$$ where
• • $x$ represents a variable or an unknown

• • a,b and c are constants

• • a ≠ 0

Here are some examples:
• $x^2 + 2x + 8 = 0$
• $4x^2 - 6x + 9 = 0$
• $x^2 - 25x = 0$
• $9x^2 + 81 = 0$

$9x + 5 = 0$ is not a quadratic equation as $x^2$ term is missing. It is a linear equation.

Few more points to remember-
• The quadratic equation involves only one unknown ($x$ in this case).
• The quadratic equation only contains powers of $x$ that are non-negative.
• The solution to the quadratic equation are those values of $x$ which make its value 0. These are called roots of the equation.

A quadratic equation with real or complex coefficients has two solutions, called roots.
These two solutions may or may not be distinct, and they may or may not be real.

One way to find the roots of a quadratic equation is by a method called 'Completing the square'.
Starting with a quadratic equation in a standard form $ax^2 + bx + c = 0$

• Divide each side by $a$
⇒ $x^2 + \frac{bx}{a} + \frac{c}{a} = 0$

• Subtract constant $\frac{c}{a}$ from both sides
⇒ $x^2 + \frac{bx}{a} = \frac{-c}{a}$

• Add the square of one-half of $\frac{b}{a}$, the coefficient of $x$, to both sides.
⇒ $x^2 + \frac{bx}{a} + \frac{b^2}{4a^2} = \frac{-c}{a} + \frac{b^2}{4a}$

• Now we can re-write left side as a square and simplify right hand side a bit.
⇒ $(x + \frac{b}{2a})^2 = \frac{b^2 - 4ac}{4a}$

• Take square root on both sides.
⇒ $x + \frac{b}{2a} = ± \frac{\sqrt{b^2 - 4ac}}{2a}$
⇒ $x = ± \frac{\sqrt{b^2 - 4ac}}{2a} - \frac{b}{2a}$

$$x = \frac{\text{-b ±\sqrt{b^2 - 4ac}}}{2a}$$ The solution(s) to a quadratic equation can be found by using this formula.

Notice the ± symbol. This means that both $\frac{\text{-b +$\sqrt{b^2 - 4ac}$}}{2a}$ and $\frac{\text{-b -$\sqrt{b^2 - 4ac}$}}{2a}$ are roots of the quadratic equation.

Let us now solve the following quadratic equation using this method.

$4x^2 - 12x + 3 = 0$

• Divide each side by 4
⇒ $x^2 - \frac{12x}{4} + \frac{3}{4} = 0$

• Subtract constant $\frac{3}{4}$ from both sides
⇒ $x^2 - \frac{12x}{4} = \frac{-3}{4}$
⇒ $x^2 - 3x = \frac{-3}{4}$

• Add the square of one-half of $3x$, the coefficient of $x$, to both sides.
⇒ $x^2 -3x + \frac{9}{4} = \frac{-3}{4} + \frac{9}{4}$

• Now we can re-write left side as a square and simplify right hand side a bit.
⇒ $(x - \frac{3}{2})^2 = \frac{6}{4}$

• Take square root on both sides.
⇒ $x - \frac{3}{2} = ± \frac{\sqrt{6}}{2}$
⇒ $x = ± \frac{\sqrt{6}}{2} + \frac{3}{2}$
⇒ $x = \frac{3 ± \sqrt{6}}{2}$
⇒ $x = \text{2.72474487 or 0.27525512}$

### Geometric Interpretation

The graph of a quadratic equation has a shape called parabola.
The values of a, b and c determine the location and size of the parabola.

See how the graphs of quadratic equation and linear equation differ from each other.
 Graphical representation of Quadratic Equation Graphical representation of Linear Equation

The simplest quadratic equation is $f(x) = x^2$. This equation represents a parabola.
In this graphical representation, $f(x) = x^2$. In this case, a = 1.

As we change the value of a, see how the shape of parabola changes.
$g(x) = \frac{1}{2}x$ ⇒ For values of $a$ less than 1, the parabola tends to be flatter and it expands outwards.
$h(x) = 2x$ ⇒ For values of $a$ greater than 1,the parabola curves inwards.
$j(x) = - x^2$. ⇒ For values of $a$ less than 0, the parabola flips upside down.