In this article, you will learn what are quadratic equations, the properties of their roots, how to represent them graphically, and how to solve them using the three methods with examples. So, let us get started.

What are Quadratic Equations?

The standard form of a quadratic equation is given below:

ax^2 + bx + c = 0, where a \neq 0

Here a is the leading coefficient of the equation and it should not be zero. The values of x that make the equation true are known as zeroes or roots of a quadratic equation. You can see that the quadratic equation is a second degree polynomial which means that the highest degree of this equation is 2.

Properties of Roots of a Quadratic equation

The roots or zeroes of a quadratic function has the following properties:
  • If c = 0, then one of the roots of a quadratic function is zero and the other is equal to \frac{-b}{a}
  • If b and c are equal to zero, then both the roots of a quadratic function are zero
  • If a is equal to c, then both the roots of the function are reciprocal to each other

Graphical Representation of the Quadratic equations

A U-shaped curve known as parabola is used to represent quadratic functions graphically. Some properties of the graphs of the quadratic functions are given below:
  • The sign of the leading coefficient "a" determines the shape of the graph of quadratic functions. If the value of "a" is greater than 0, i.e. it is positive, then the parabola opens upwards.
  • If the leading coefficient "a" is less than 0, i.e. it is negative, then the parabola opens downwards.
  • The maximum or minimum (depending upon the shape of the parabola) point of the parabola is known as a vertex. A line that vertically passes through the vertex is known as the axis of symmetry.
  • The points where the parabola intersects the x-axis are known as x-intercepts of the function. These x-intercepts are the roots or zeros of the function.

Methods for Solving Quadratic Equations

Three methods are used to solve the quadratic equations. These methods are factoring, completing the square and quadratic formula

i. Factoring

It is the simplest way to find the roots of the quadratic functions. All we have to do is expand the middle term to get four terms, group the like terms and factor them. The following steps are generally involved in this process:
  • All the terms are taken on the left-hand side of the equation. The equation is set to zero.
  • Expand the middle term into two terms in such a way that the product of two terms is equal to ax^2c and the sum or difference of two terms is equal to bx.
  • Group the like terms and find factors.
  • Set each factor equal to zero and solve the new equations for the unknown variable x.
  • You can also substitute the value of x in the original equation to check whether it satisfies the equation or not.

Example

Solve x^2 + 18x = - 80 by factoring.

Solution

You can see that the constant is on the right hand side of the equation. Hence, we will bring it to the left hand side of the equation and set the equation equal to zero.

x^2 + 18x + 80 = 0

Now, we will expand the middle term into two terms in such a way that the product is equal to 80x^2 and sum or difference is equal to 18x

x^2 + 10x + 8x + 80 = 0

x (x + 10) + 8 (x + 10) = 0

(x + 10) (x + 8) = 0

We got two new equations by setting each factor equal to zero. We will solve the two equations separately for x:

(x + 10) = 0 and (x + 8) = 0

x = - 10 and x = - 8

Hence -10 and -8 are the roots of the equation. To check our answer, we will substitute these values in the original equation:

x^2 + 18x = - 80 at x = -10

(-10)^2 + 18 (-10) = -80

100 - 180 = -80

-80 = -80

x^2 + 18x = -80 at x = -8

(-8)^2 + 18 (-8) = -80

64 - 144 = -80

-80 = -80

Hence, it is proved that our answer is correct, i.e, -10 and -8 are the solutions to the equation.

 

ii. The Quadratic Formula

Most of the time we cannot find factors of a quadratic equation. In these problems, using the quadratic formula helps us to determine the roots or zeroes of the equation. We especially use the quadratic formula when the roots or zeroes of the quadratic equations are not rational numbers. The following formula is used for a quadratic equation ax^2 + bx + c = 0,

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

The element of the formula present under the radical sign b^2 - 4ac is called the discriminant. The discriminant of the quadratic formula determines the following three possibilities:

  • Two different roots are obtained if the discriminant b^2 - 4ac is a positive number.
  • A function has only one real root, zero or solution if the discriminant b^2 - 4ac is zero.
  • A function has no real solution if the discriminant b^2 - 4ac is less than 0, i.e. it is a negative number.

Example

Find the roots of the equation 7x^2 - 2x - 8 = 0

Solution

We cannot factorize the equation in this example, therefore we will move forward to use the quadratic formula to find the roots of the equation:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Here, a = 7, b = -2 and c = -8.

We will substitute the values of a, b and c in the formula:

x = \frac{- (-2) \pm \sqrt{(-2)^2 - 4(7) (-8)}}{2(7)}

x = \frac{2 \pm \sqrt{4 - - 224}}{14}

x = \frac{2 \pm \sqrt{228}}{14}

x = \frac{2}{14} \pm \frac{\sqrt{57}} {7}

Simplify the above expression to get the following solutions:

x = 1.2214 and x = -0.935691

 

iii. Completing the Square

The third method of solving the quadratic equations is completing the square method. If the roots of the equation are real of imaginary numbers, then this method is the most suitable one. The following steps are involved in solving the quadratic equations through this method:

  • Take constant on the right-hand side of the equation to convert the equation into the form ax^2 + bx = -c
  • The leading coefficient "a" should be equal to 1. If it is not the case then you should make it equal to 1 by dividing the entire equation by the leading coefficient a.
  • Use the value of b to make a perfect square on the left-hand side of the equation. Add (\frac{b}{a})^2 to both sides of the equation.
  • In the last step, take the square root on both sides of the equation and solve the resulting equation for unknown value x.

Example

Solve the equation x^2 - 12x + 9 = 0 using completing the square method.

Solution

Take the constant on the right hand side of the equation:

x^2 - 12x = -9

Calculate the value of (\frac{b}{a})^2 and add it to both sides of the equation:

(\frac{-12} {2})^2 = 36

x^2 - 12x + 36 = - 9 + 36

x^2 - 12x + 36 = (x - 6)^2

(x - 6)^2 = 25

Take square root of both sides of the equation and solve the resulting equation for x:

\sqrt {(x - 6)^2} = \sqrt{25}

x - 6 = \pm 5

x - 6 + 5 = 0 or x - 6 - 5 = 0

x - 1 = 0 or x - 11 = 0

x = 1 or x = 11

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.