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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:
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.
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
We can solve the quadratic equation to find its roots in different ways.
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) =
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) =
=
=
=
= {-3, -2}
Thus, the two roots of the quadratic equation x2 + 5x + 6 = 0 are {-3, -2}
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}
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
Here, b =
=>
Step 3: Factor the right side of the equation into a perfect square
=>
Step 4: Rewriting the equation in terms of y
=>
When the quadratic function f(x) = y = 0
=>
=>
=>
=> x= {-3, -2}
Thus, the two roots of the quadratic equation x2 + 5x + 6 = 0 are {-3, -2}
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)
The nature of the roots of the quadratic equation depends on the value of the discriminant as follows:
Find the value(s) of g if the quadratic equation 3y2 + gy + 2 = 0 has equal roots.
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 =
Thus, when the given quadratic equation has equal roots, g =
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 =
Derivation
α + β =
=
=
The product of the roots of a quadratic equation is equal to the constant term divided by the leading coefficient.
α.β =
=
=
=
Find the sum and the product of the roots of the quadratic equation a2 – 5a – 14 = 0
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 =
= 1
Thus the sum of the roots of the quadratic equation a2 – 5a – 14 = 0 is 1
α.β =
= -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}
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