Last modified on August 3rd, 2023

chapter outline

 

Roots of Quadratic Equation

The roots of a quadratic equation are the values of the variable that satisfies the equation. They are also known as the ‘zeroes’ of the quadratic equation. For the equation ax2 + bx + c = 0 the two roots α and β are:

  • α=b+b24ac2a
  • β = bb24ac2a

For example, the two roots of the quadratic equation x2 + 5x + 6 = 0 are {-3, -2} because they satisfy the equation when their values are substituted.

When x = -3, (-3)2 + 5 × -(3) + 6 = 0

When x = -2, (-2)2 + 5 × -(2) + 6 = 0

Without solving the equation 3x2 – 2x – 1 = 0, find whether x = 1 is a root or solution of this equation.

Solution:

Substituting x = 1 in the given equation 3x2 – 2x – 1 = 0, we get
3(1)2 – 2 (1) – 1 = 0
=> 3 – 2 – 1 = 0
=> 3 – 3 = 0
Therefore, x = 1 is a solution of the given equation 3x2 – 2x – 1 = 0

How to Find the Roots of a Quadratic Equation

We can solve the quadratic equation to find its roots in different ways.

Using the Quadratic Formula

As we know α and β are the two roots of the quadratic equation, whose values can be determined using the quadratic formula:

(α, β) = b±b24ac2a

Let us verify if, indeed, the roots of the quadratic equation x2 + 5x + 6 = 0 are {-3, -2}

In the equation x2 + 5x + 6 = 0, a = 1, b = 5, c = 6

(α, β) = 5±(5)24×1×62×1

= 5±25242

= 5+12

= 5+12 and 512

= {-3, -2}

Thus, the two roots of the quadratic equation x2 + 5x + 6 = 0 are {-3, -2}

By Factoring

Here we will factor the middle term ‘5x’ of the same equation x2 + 5x + 6 = 0

x2 + 5x + 6 = 0

=> x2 + 3x +2x + 6 = 0

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

=> (x + 3)(x + 2)=  0

=> x + 3 = 0 or x + 2 = 0

=> x = {-3, -2}

Thus, the two roots of the quadratic equation x2 + 5x + 6 = 0 are {-3, -2}

By Completing the Square

Step 1: Isolating the x2 and x terms to one side of the equation

y = x2 + 5x + 6

=> y – 6 = x2 + 5x

Step 2: Add (b2)2 to both sides of the equation

Here, b = (52)2 = 25/4

=> y6+254=x2+5x+254

Step 3: Factor the right side of the equation into a perfect square

=> y+14=(x+52)2

Step 4: Rewriting the equation in terms of y

=> y=(x+52)214

When the quadratic function f(x) = y = 0

(x+52)214=0

=> (x+52)2=14

=> x+52=±14

=> x={+1252,1252}

=> x= {-3, -2}

Thus, the two roots of the quadratic equation x2 + 5x + 6 = 0 are {-3, -2}

By Graphing

Graphing the quadratic equation manually using a graphing calculator and then identifying the x-intercepts will give the roots of the quadratic equation.

The graph shows the two x-intercepts are (-2, 0) and (-3, 0). Thus the two roots of the quadratic equation are (-3, -2)

Nature of Roots of the Quadratic Equation

The nature of the roots of the quadratic equation depends on the value of the discriminant as follows:

  • If b2 – 4ac > 0, the quadratic equation has 2 real solutions
  • If b2 – 4ac < 0, the quadratic equation has no real solutions but two different complex or imaginary roots
  • If b2 – 4ac = 0, the quadratic equation has one real root

Find the value(s) of g if the quadratic equation 3y2 + gy + 2 = 0 has equal roots.

Solution:

As we know, thequadratic equation ax2 + bx + c = 0 has equal roots if its discriminant b2 – 4ac = 0
Here, a = 3, b = g, and c = 2
Putting the value in the discriminant, we get
b2 – 4ac = 0
=> (g)2 – 4 × 3 × 2 = 0
=> g2– 24 = 0
=> g2 = 24
=>g = ±26
Thus, when the given quadratic equation has equal roots, g = 26 or g = 26

Sum of the Roots

The sum of the roots of a quadratic equation is equal to the negation of the coefficient of the second term, divided by the leading coefficient. 

α + β = ba

Derivation

α + β = b+b24ac2a+bb24ac2a

= b+b24acbb24ac2a

= 2b2a = ba

Product of the Roots

The product of the roots of a quadratic equation is equal to the constant term divided by the leading coefficient.

α.β = b+b24ac2a×bb24ac2a

= b2+bb214acbb24ac(b24ac)4a2

= b2b2+4ac4a2

 = 4ac4a2 = ca

Find the sum and the product of the roots of the quadratic equation a2 – 5a – 14 = 0

Solution:

Comparing the given quadratic equation with the standard form ax2 + bx + c = 0, here a = 1, b = -5, c = -14
As we know,
α + β = ba
= 1
Thus the sum of the roots of the quadratic equation a2 – 5a – 14 = 0 is 1
α.β = ca
= -14
Thus the product of the roots of the quadratic equation a2 – 5a – 14 = 0 is -14

Write the quadratic equation with the given roots {7, -2}

Solution:

The quadratic equation can be obtained by multiplying the factors formed by the roots {7, -2}
The factors are (x – 3)(x + 2)
Thus, the quadratic equation is (x – 3)(x + 2) = 0
=> x2 + 2x – 3x – 6 = 0
=> x2 – x – 6 = 0

Last modified on August 3rd, 2023

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