Quadratic Equation

A quadratic equation is an equation of the form
\( ax^2+bx+c=0\)
where the leading coefficient \(a \ne 0\).

A quadratic equation (from the Latin quadratus for "square") is an equation that can be rearranged in standard form as
\( ax^2+bx+c=0\)
where x represents an unknown, and a, b, and c represent known numbers, where \( a \ne 0\). If a = 0, then the equation is linear, not quadratic, as there is no \(ax^2 \)term.

1. The standard form of Quadratic Equation is
   \( ax^2 + bx + c = 0\) where \(a \ne 0\)
2. Because the quadratic equation involves only one unknown, it is called "univariate".
3. The numbers a, b, and c are the coefficients, they are also respectively called quadratic coefficient, linear coefficient, and the constant or free term.
4. The quadratic equation contains only powers of x that are non-negative integers, and therefore it is a polynomial equation. In particular, it is a second-degree polynomial equation, since the greatest power is two.
5. Equations such as \( x^2 = 64, x^2 -5x = 0\), and \(x^2 + 4x = 5\) are called quadratic equations. This is because in each of these equations, the greatest exponent of the variable x is 2.
6. A quadratic equation can be factored into an equivalent equation \( ax^2+bx+c=(x-\alpha)(x-\beta)=0\)
   where \(\alpha\) and \(\beta\) are the solutions for x.
7. The values of x that satisfy the equation are called solutions, roots or zeros.
   A quadratic equation has at most two solutions. If there is only one solution, it is a double root.
   A quadratic equation always has two roots, if complex roots are included and a double root is counted for two.
8. A quadratic polynomial has a parabola as its graph

Three forms of Quaadratic Equation

We can obtain three different representations of a quadratic function

1. Its standard representation: \(f(x) = ax^2 + bx + c\).
   The function f has two zeros, a double zero, or no zero according to whether its discriminant \(b^2-4ac\) is positive, zero, or negative, respectively
2. Its representation in vertex form: \( f (x) = a(x-p)^2 + q\). Here, the point \((p, q)\) is called the vertex of the graph.
   The graph of f(x) has bilateral symmetry with respect to the vertical line defined by \(x = p\).
   If a > 0, f(x) is decreasing on (−∞, p] and therefore increasing on [p,∞).
   If a < 0, then f(x) is increasing on (−∞, p] and therefore decreasing on [p,∞).
   If a > 0, then f(x) achieves its minimum at p.
   If a < 0, then f(x) achieves its maximum at p.
3. Its representation in factored form: \(f (x) = a(x−\alpha)(x−\beta)\), where \(\alpha, \beta\) are the zeros of f(x).Also \(\alpha+ \beta=-\frac{b}{a}\) and \(\alpha \beta=\frac{c}{a}\)

Each of the three representations of a quadratic function reveals a different facet of the function. The quadratic formula is expressed in terms of (1), for standard notion, and (2) displays the line of symmetry of the graph of f(x) and also where it achieves its maximum or minimum. If the zeros of f(x) are our main interest, then (3) displays its zeros explicitly. Together, the three representations give a well-rounded picture of f(x); none gets it done alone.

Three forms of Quaadratic Equation

A solution of a quadratic equation is also called a root of the equation. The intuitive meaning of the root of a quadratic equation can be given pictorially, as follows.

The graph of y = 2x² − x − 3 intersects the x-axis at (−1, 0) and (1.5, 0), and a simple computation confirms that −1 and 1.5 are roots of 2x² − x − 3 = 0

When solving quadratic equations, we can use two methods:

1. Factoring
2. Quadratic Formula
   Not every quadratic equation can be solved by factoring. In this case, we need to use the quadratic formula.
   The Quadratic Formula is
   \( x= \frac{-b \pm \sqrt {b^2-4ac}}{2a} \)

Proof of Quadratic Formula

Prove that two roots of quadratic equations \(ax^2 + bx + c = 0\) are \(\frac{-b - \sqrt{b^2-4ac}}{2a}\) and \(\frac{-b + \sqrt{b^2-4ac}}{2a}\)
Proof
Given quadratic equation is
\(ax^2 + bx + c = 0\)
or \(ax^2 + bx =-c\)
Dividing both sides by a, we get
\(x^2 + \frac{b}{a}x = - \frac{c}{a}\)
Adding \(\frac{b^2}{4a^2}\) on both sides we get
\(x^2 + \frac{b}{a}x + \frac{b^2}{4a^2}= \frac{b^2}{4a^2}- \frac{c}{a}\)
or \(\left( x+\frac{b}{2a} \right )^2=\frac{b^2-4ac}{4a^2}\)
Taking square roots we get
\(x+\frac{b}{2a}=\frac{\pm \sqrt{b^2-4ac}}{2a}\)
or \(x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\)
So, the roots of quadratic equations \(ax^2 + bx + c = 0\) are \(\frac{-b - \sqrt{b^2-4ac}}{2a}\) and \(\frac{-b + \sqrt{b^2-4ac}}{2a}\)

To solve a Quadratic equation using the formula, we use the following steps:

1. Put the quadratic equation into standard form : \(ax^2 + bx + c = 0\)
2. Write out the value for a, b, and c
3. Substitute value in the formula, solve and get the roots
4. Check each root in the original equation

Nature of Roots

In the quadratic equation \(ax^2+bx+c=0\), the expression \(b^2-4ac\) is called the discriminant, and is often represented using an upper case D or an upper case Greek delta \( \Delta \) given as
\( \Delta=b^2-4ac \)
A quadratic equation with real coefficients can have either one or two distinct real roots, or two distinct complex roots. In this case, the discriminant determines the number and nature of the roots.

Quadratic Equation

Quadratic Equation

Quadratic Equation

Δ is positive

Δ is zero

Δ is negative

This figure given above plots three quadratic equations to illustrate the effects of discriminant values. There are three cases:

1. When the discriminant \( \Delta\) is positive, the curve of quadratic equation (parabola) intersects the x-axis at two points. Then there are two distinct roots
   \( \frac {-b+\sqrt{ \Delta }}{2a}\) and \( \frac {-b-\sqrt{ \Delta }}{2a}\)
   both of which are real numbers.
   For quadratic equations with rational coefficients,
   1. if the discriminant is a square number, then the roots are rational
   2. if the discriminant is a NOT square number, then the roots are irrational
      This kinds of irrational roots always occur in conjugate pairs, which are of the form
      \( \alpha= \frac {-b+\sqrt{ b^2-4ac }}{2a}\) and \( \beta= \frac {-b-\sqrt{ b^2-4ac }}{2a}\)
      or \( \alpha= \frac {-b}{2a}+\frac{\sqrt{ b^2-4ac }}{2a}\) and \( \beta= \frac {-b}{2a}-\frac{\sqrt{ b^2-4ac }}{2a}\)
      or \( \alpha= p+\sqrt{q}\) and \( \beta= p-\sqrt{q}\)
2. When the discriminant \( \Delta\) is zero, the vertex of the curve of quadratic equation (parabola) touches the x-axis at a single point. Then there is exactly one real root
   \( \frac{-b}{2a}\)
   sometimes called a repeated or double root.
3. When the discriminant \( \Delta\) is negative, the curve of quadratic equation (parabola) does not intersect the x-axis at all. Then there are no real roots. Rather, there are two distinct (non-real) complex roots
   This kinds of complex roots always occur in conjugate pairs, which are of the form
   \( \alpha= \frac {-b+\sqrt{ b^2-4ac }}{2a}\) and \( \beta= \frac {-b-\sqrt{ b^2-4ac }}{2a}\)
   or \( \alpha= \frac {-b}{2a}+\frac{\sqrt{ b^2-4ac }}{2a}\) and \( \beta= \frac {-b}{2a}-\frac{\sqrt{ b^2-4ac }}{2a}\)
   or \( \alpha= p+i q\) and \( \beta= p-iq\)
   In these expressions i is the imaginary unit.
4. Thus the roots are distinct if and only if the discriminant is non-zero, and the roots are real if and only if the discriminant is non-negative.

Based on the discriminant \( b^2-4ac\) of the quadratic equation \( ax^2+bx+c=0\) , the corresponding solution will be

If the discriminant is	then there (roots)
positive \( b^2-4ac >0\) and a perfect square	are two rational roots
positive \( b^2-4ac >0\) and NOT a perfect square	are two irrational roots
zero: \( b^2-4ac=0\)	is one rational roots
negative: \( b^2-4ac < 0\)	are two complex roots