## Quadratic equations, Quadratic Functions

Many** IB Maths Tutors** consider **quadratic equations** as a very important topic of maths. There are the following ways to solve a quadratic equation

**► Factorization method**

**►complete square method**

**► graphical method**

**► Quadratic formula method**

**Quadratic equations, Quadratic Functions, and Quadratic Formula**

The quadratic formula is the strongest method to solve a quadratic equation. In this article, I will use a few steps to prove** the quadratic formula.**

Given equation: ax²+bx+c=0

**Step-1:** transfer constant term to the right side

ax²+bx=-c

**Step-2:** divide both sides by coefficient of x²

**Step-3:**write (coefficient of x/2)² that is (b/2a)²=b²/4a²

**Step-4:**Add this value to both sides

x+b/2=±√b²-4ac/a²

** **

This formula is known as the quadratic formula. We have used a simple way to **prove quadratic formula** we can put values of a, b and c from any equation and find the value of x (the variable) by directly using this formula.

**a+√b**then the second root will automatically become a-√b. that can be understood easily as we use one +ve and one -ve sign in the quadratic formula.

These types of roots are called

**Conjugate Roots**.

**Nature Of Roots:**

^{2}+ bx + c = 0 where a, b, c Q & then

^{2}– (a + b) x + a b = 0 i.e.

**x**

^{2}– (sum of roots) x + product of roots = 0**Common Roots Of 2 Quadratic Equations [Only One Common Root]-**Let

be the common root of ax² + bx + c = 0 & a’x

^{2}+ b’x + c’ = 0

Therefore

Every pair of the quadratic equation whose coefficients fulfills the above condition will have one root in common.

**The condition that a quadratic function-**

**Reducible Quadratic Equations-**These are the equations which are not quadratic in their initial condition but after some calculations, we can reduce them into quadratic equations

i.e., , , , all these equations can easily be reduced into quadratic equations by applying the method of substitution.

**Example-**Solve this equation and find x

** **Ans:

let y= then given equation will become

it’s a simple quadratic equation we can be easily factorized it and solve so y=–3,2

so =-3

the final equation can be solved using Quadratic formula and the same process can be repeated for y=2

(ii) If a variable is added with it is own reciprocal, then we get a quadratic equation i.e, all these equations can be reduced into quadratic by replacing one term by any other variable.

**Standard Form of a Quadratic Function**-A quadratic function y=ax^{2}+ bx + c can be

reduced into standard form by the method of completing the square. If we

draw the graph of this function we shall get a parabola with vertex (h,k). The parabola will be upward for a>0 and downward for a<0

**Maximum and Minimum value of a quadratic function- **If the function is in the form

Then ‘h’ is the input value of the function while ‘k’ is its output.

(i) If a>0 (in case of the upward parabola) the minimum value of f is f(h)=k

f(x) has a minimum value at vertex if a>0 and at

f(x) has a maximum value at vertex if a<0 and at

In the next post about quadratics, I shall discuss discriminant, nature of roots, relationships between the roots. In the meantime, you can download the pdf and solve practice questions