Quadratic Equation: Definition, Methods of Solving, Roots, Formulas

A quadratic equation is a second-degree polynomial equation in the form ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0where aaa, bbb, and ccc are constants, and xxx represents the variable. The term "quadratic" comes from "quad," meaning square, because the highest power of the variable is squared (i.e., x2x^2x2).

What is Quadratic Equation?

Also Check: Isosceles Triangle

Methods of Solving Quadratic Equations

1. Factoring Method

• First, write the equation in standard form: ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0.
• Next, factor the quadratic expression into two binomials, such as (px+q)(rx+s)=0(px + q)(rx + s) = 0(px+q)(rx+s)=0.
• Solve for xxx by setting each factor equal to zero and solving for the variable.

2. Quadratic Formula

x=2a−b±b2−4ac​​

• If b2−4ac>0b^2 - 4ac > 0b2−4ac>0, there are two distinct real roots.
• If b2−4ac=0b^2 - 4ac = 0b2−4ac=0, there is exactly one real root (or two equal real roots).
• If b2−4ac<0b^2 - 4ac < 0b2−4ac<0, the roots are imaginary or complex.

3. Completing the Square

• Start with the equation in standard form ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0.
• Divide the equation by aaa (if a≠1a \neq 1a=1) to normalize the coefficient of x2x^2x2.
• Move the constant term c/ac/ac/a to the other side of the equation.
• Add the square of half the coefficient of xxx to both sides of the equation.
• Factor the left-hand side, and then solve for xxx by taking the square root of both sides.

4. Graphing Method

• If the parabola crosses the x-axis at two points, there are two real solutions.
• If it touches the x-axis at one point, there is one real solution.
• If the parabola does not intersect the x-axis, the equation has no real solutions.

5. Using the Square Root Property

• Isolate x2x^2x2 on one side of the equation.
• Take the square root of both sides.
• Solve for xxx, keeping in mind that there are both positive and negative roots.

Also Check: Shapes

Roots of a Quadratic Equation

• For x=−1x = -1x=−1:
• −3(−1)−4=1+3−4=0
• For x=4x = 4x=4:
• (4)2−3(4)−4=16−12−4=0

Nature of Roots of the Quadratic Equation

1. If D>0D > 0D>0 , the roots are real and distinct.
2. If D=0D = 0D=0 , the roots are real and equal.
3. If D<0D < 0D<0 , the roots are imaginary or complex , meaning no real solutions exist.

Also Check: Surface Area of a Hemisphere

Formulas Related to Quadratic Equations

• If 𝐷>0, the equation has real and distinct roots.
• If 𝐷=0, the equation has real and equal roots.
• If 𝐷<0, real roots do not exist, and the roots are imaginary.
• The formula for finding the roots of the quadratic equation is: 𝑥=−𝑏±𝑏2−4𝑎𝑐2𝑎
• The sum of the roots of the equation is 𝛼+𝛽=−𝑏𝑎
• The product of the roots is 𝛼𝛽=𝑐𝑎
• The quadratic equation whose roots are 𝛼 and 𝛽

𝑥2−(𝛼+𝛽)𝑥+𝛼𝛽=0

(𝑎1𝑏2−𝑎2𝑏1)

(𝑏1𝑐2−𝑏2𝑐1)=(𝑎2𝑐1−𝑎1𝑐2)2

When , 𝑎>0, the quadratic function

𝑥=−𝑏2𝑎

𝑥=−𝑏2𝑎x=− 2ab