Quadratic Equation

Introduction to Quadratic Equation

The topic of quadratic equation in mathematics is very important and helps us solve many real-life problems. The word “quadratic” comes from the Latin word “quadratus”, which means “square”. So a quadratic equation is an equation that has a power of 2 in the variable.

A quadratic equation is written in the form  ax2+bx+c=0, where A, B, and C are numbers, and a is not equal to zero. For example, x2+5x+6=0 is a quadratic equation. In such equations, the highest power of variables (x) is always 2. There can be two solutions (called roots) in quadratic equations, which can be real or imaginary.

In this article, we will provide more information about quadratic equations, their standard forms, methods to solve them, characteristics, and applications in real life. With the solved examples, this step-by-step guide will help you gain a clear and simple understanding of quadric equations.

 

Table of Contents:

• Quadratic Equation Definition
• Quadratic formula
• Formula for Solving Quadratic Equations
• Roots of Quadratic Equation
• Nature of Roots
• Common Roots
• Solving Quadratic Equations
• Factorization Method
• Using Formula Method
• Completing the Square Method
• Graphical Solution
• Range of Quadratic Equation
• Maximum and Minimum Value
• Sign Convention
• Quadratic Equations in Two Variables
• Biquadratic Equation
• Quadratic Functions in the Real World
• Conclusion
• FAQs on Quadratic Equation

Quadratic Equation Definition

A quadratic equation is a special type of algebraic equation where the highest power of the variable is 2. This means that the equation will always have a word like x2. When we write a quadratic polynomial and make it equal to zero, it becomes a quadratic equation.

The general form of a quadric equation is:

ax2+bx+c=0

Here it is, a, b, c are numbers, and a ≠ 0 (because if a = 0, it will no longer be quadratic anymore).

Some examples are:

• 3x2+x+5=0

• −x2+7x+5=0

• x2+x=0

The value of x that makes the equation true is called the roots or solutions of the quadratic equation.

Quadratic formula

• To find the roots of a quadratic equation, we use a formula called the quadratic formula. It's:

x=[−b±(b2)−4ac]/2a

• This formula helps us find two possible values ​​of x (because "±" means plus or minus).

 

Formula for Solving Quadratic Equations

1. Roots of a quadratic equation

• For a quadratic equation ax2+bx+c=0, the roots are found using the formula:

• x=(−b±D)/2a, where D=b2−4ac

• Here, D is called is discriminant.

2. Nature of roots

• If D > 0, the Roots are real and different.

• If D = 0, roots are real and equal.

• If D < 0, the roots are not real.

3. Root as Pairs

• If the roots are in the form α+iβ and α−iβ, they are called a conjugate pair.

4. Sum and product of roots

• If soots are α and β, then:

• Sum (α+β)=−b/a

• Product (α×β)=c/a

5. Writing quadratic equation using roots

• If roots are α and β, then the equation can be written as:

•  x2−(α+β)x+(αβ)=0

6. Common roots in 2 equations

• For 2 quadratic equations ax2+b1x+c1=0 and a2x2+b2x+c2=0:

• They have 1 root in common if a special relation between coefficients is true.

• They have both roots in common if a1/a2=b1/b2=c1/c2. 

7. Maximin and minimum value

• For ax2+bx+c=0,

• If a>0, the parabola opens upwards, and it has a minimum value at x=−b/2a.

• If a<0, the parabola opens downward, and it has a maximum value at x=−b/2a.

8. For a cubic equation

• If If the equation is ax2+bx+c=0 with roots α,β,γ, then

• α+β+γ=–b/a

• αβ+βγ+γα=c/a

• αβγ=–d/a

9. Identity case

• If a=b=c=0, then the equation is true for many values of x; this is called an identity.

 

Roots of Quadratic Equation

The roots of a quadratic equation are the value of x that makes the equation true. When we put root in place of x, the equation becomes zero. For example, x2−5x+6=0, the roots are 2 and 3, as both make the equation equal to zero.

Nature of Roots

The nature of roots depends on the value of D=b2−4ac:

Common Roots

When two quadratic equations share at least one root, it is called a common root.

Example:

• Equation 1: x2−5x+6=0 (root 2,3)

• Equation 2: x2−8x+15=0 (root 3,5)

• Here, the common root is 3.

 

Solving Quadratic Equations

A quadratic equation is usually written as:

ax2+bx+c=0

Where a≠0

There are 3 common methods to solve quadratic equations:

• Factorization Method

We split the middle term and factorize.

Example: Solve x2+5x+6=0

Step 1: Split 5 as 2 + 3

x2+2x+3x+6=0

Step 2: Group and factorize

x(x+2)+3(x+2)=0

(x+2)(x+3)=0

 

Step 3: Roots are

x=−2,x=−3

• Using Formula Method 
  

We  use the quadratic formula:
x=−b±b2−4ac2a

Example: Solve 2x2+3x−2=0
Here, a = 2, b=3, c= −2
x=−3±32−4(2)(−2)2(2)
x=−3±9+164
x=−3±54
So, 
x=−3+54=24=12,x=−3−54=−84=−2

• Completing the Square Method
  

We make the equation a perfect square.
Example: Solve x2+6x+5=0
Step 1: Write as x2+6x=−5
Step 2: Add (62)2=9  on both sides.

x2+6x+9=4
(x+3)2=4

Step 3: Take the square root

x+3=±2
x=−3+2=−1,x=−3−2=−5

Graphical Solution

A quadratic equation forms a parabola when drawn on a graph.

• If the parabola cuts the x-axis, then it has real roots.

• If it touches the x-axis, then it has equal roots.

• If it does not touch the x-axis, then it has no real roots.

Example: Graph of y=x2−4
It cuts the x-axis at x= −2 and x=2
So, the roots are −2,2.

Range of Quadratic Equation

The range means the values a quadratic function can take.

• If the parabola opens upward (a>0), the minimum value is at the vertex.

• If oaravola opens downward(a<0), the maximum value is at the vertex.

Formula of vertex: Substitute this in the equation to get the min/max value.

x=−b2a

Example: y=x2+4x+5

Here, a= 1, b=4.

x=−42(1)=−2

At x = −2,

y=(−22)+4(−2)+5=4−8+5=1

Maximum and Minimum Value

From the above:

• If a > 0, the parabola opens upward, and the minimum value exists here.

• If a < 0, the parabola opens downward and the maximum value exists here.

Sign Convention

While solving quadratic equations, we must be careful with the signs of a,b, and c.

• Always check positive (+) or negative (−).

• Mistakes usually happen in −b and ±.

• Example: In a formula  −b±b2−4ac2a, the −b means the opposite sign of b.

Quadratic Equations in Two Variables

A quadratic equation in 2 variables looks like:

ax2+by2+cxy+dx+ey+f=0

Examples:x2+y2=25 , this represents a circle of radius 5.

Example: x2+y2−4x−6y+9=0, this also represents a circle.

Biquadratic Equation

A biquadratic equation is of degree 4, like:

ax4+bx3+cx2+dx+e=0

It can often be sloved by substituing: z=x2

Example: Solve x4−5x2+4=0

Step 1: Let z=x2.Then,

z2−5z+4=0

Step 2: Factorize:

(z−4)(z−1)=0

So, z =4 or z = 1.

Step 3: Put back x2=z:

● If x2=4.so, x = 2 or x = −2.

● If x2=1.so, x = 1 or x = −1.

● Final answer: Roots are −2,−1,1,2.

Nature of Roots and Discriminant in Quadratics

 

Quadratic Functions in the Real World

Quadratic functions play a Major Role in real-life Life Scenarios and are usually encountered in Secondary School Algebra. Their Applications Go Beyond Classroom Theory and Make A Basis To Solve Many Practical Problems.

Applications of Quadratic Equations in Word Problems

Quadratic Equations are widely used in Word Problems involving quadratics, Especially Where Variables Change in a Square Manner. Some Common Real-World Applications Include:

• Projectile motion: Determination of the Height of an Object Over Time Makes A Porablala.

• Profit and Revenue Analysis: Maximizing or minimizing the price using the maximum and minimum price on the vertex.

• Geometry Problems: Finding Dimensions with the Obstacles of the Given Area.

• Engineering and Design: Creating Curved Paths or Structures Based on Quadratic Functions.

• Physics Calculation: Calculation of Force, Speed​​, or Energy that includes a square term.

Use of Quadratics in Mathematics Curriculum

Quadratic Concepts are Integral Parts of the Mathematics curriculum and are introduced in Secondary school algebra.

• Strengthen understanding of Algebra, Polynomial Equations, and Graphing Skills.

• Teach Students How to Solve Quadratic Equations Using Several Methods.

• Explain Real and Complex Roots, the Nature of Roots, and the Discriminant of a quadratic.

• Create Basic Knowledge of Calculus, Optimization, and Analytical Logic.

• Encourage Visual Learning through the graph of Quadratic Equations, Focus on Axis With Symmetry, Vertex Form, and intersection X-axis.

 

Conclusion

Quadratic equations are very important in math because they teach us how to solve problems with numbers raised to the power of 2. We can solve them using methods like factorization, completing the square, or the quadratic formula, and check the type of root with the discriminant. Graphs of quadratics help us see parabolas, symmetry, and maximum or minimum values. Beyond studied quadratic equations are used in real life to solve problems in physics, business, and engineering, such as finding heights, profits, or distances.

 

FAQs on Quadratic Equation

1. How do you explain quadratics?

Ans: In math, we define a quadratic equation as an equation of degree 2, meaning that the highest exponent of this function is 2. The standard form of a quadratic is y=ax2+bx+c, where a, b, and c are numbers and a cannot be 0. Examples of quadratic equations include all of these: y=x2+3x+1.

2. What math is quadratics in?

Ans: Quadratic Equation. Quadratic equations are second-degree algebraic expressions and are of the form ax2+bx+c=0. The term "quadratic" comes from the Latin word "quadratus," meaning square, which refers to the fact that the variable x is squared in the equation.

3. Why is quadratic 2 and not 4?

Ans: The prefix quad means “four,” and quadratic expressions involve powers of x up to the second power, not the fourth power.

4. Are quadratics algebra 1 or 2?

Ans: Algebra 1 typically includes evaluating expressions, writing equations, graphing functions, solving quadratics, and understanding inequalities.

5. How to derive the quadratic formula?

Ans: The quadratic formula, x=−b±b2−4ac2a, can be derived by completing the square on the general standard form of a quadratic equation. Recall that completing the square is a method for solving quadratic equations.