A quadratic equation is an equation which is of the form ax2 + bx + c = 0 where a, b, and c are real numbers and a is not equal to zero.
Since it is a second degree equation, it has two solutions. The solutions of a quadratic equation are called the roots of the quadratic equation.
We can use the following methods to solve quadratic equations:-
a) Solve Quadratic Equations by Factoring.
b) Solve Quadratic Equations by Finding Square Roots.
c) Solve Quadratic Equations by Completing the Square.
d) Solve Quadratic Equations using Quadratic Formula.
e) Solve Quadratic Equations by Graphing.
We can solve the quadratic equations by using factoring method.
The general form of a quadratic equation is,
x2 - (sum of the roots) x + product of the roots = 0
· To factor a quadratic equation which is of the form, x2 + bx + c, find two real numbers m and n such that,
mn = c and m + n = b
If we find two integers m and n with the above conditions, then we can write x2 + bx + c as (x + m)(x + n) and then use zero product property to find the solution.
· If the quadratic equation is of the form ax2 + bx + c, then find two real numbers m and n such that,
mn = ac and m + n = b
If we find the two integers m and n with the above conditions, then we can write ax2 + bx + c as ax2 + mx + nx + c and then go ahead with factor by grouping method.
Let us solve a few examples to understand factoring method.
Example 1:
Solve x2 - x - 30 = 0 by factoring method.
Solution:
Step 1:
The given quadratic equation is of the form x2 + bx + c = 0, where b = -1 and c = -30.
Now, find two real numbers m and n such that, m + n = -1 and mn = -30.
The two numbers are 5 and -6.
Step 2:
Sum of the roots = 5 + (-6)
= 5 - 6
= -1
Step 3:
Product of the roots = (5)(-6)
= -30
Step 4:
So, we can write x2 - x - 30 as (x + 5) and (x - 6).
(x + 5)(x - 6) = 0
Now, use zero product property and simplify.
(x + 5)(x - 6) = 0
x + 5 = 0 or x - 6 = 0
x = -5 or x = 6
Step 5:
So, the solution set is {-5, 6}.
Example 2:
Solve 2x2 + x - 45 = 0 by factoring method.
Solution:
Step 1:
The given quadratic equation is 2x2 + x - 45 = 0, which is of the form ax2 + bx + c = 0, where a = 2, b = 1, and c = -45.
Now, find two real numbers m and n such that, mn = ac = (2)(-45) = -90 and m + n = 1.
The two numbers are 10 and -9.
Step 2:
Sum of the roots = 10 + (-9)
= 10 - 9
= 1
Step 3:
Product of the roots = (10)(-9)
= -90
Step 4:
Rewrite the given equation in the form ax2 + mx + nx + c as
2x2 + 10x - 9x - 45 = 0
Group the first two terms and the last two terms and then take the common factors out.
(2x2 + 10x) - (9x + 45) = 0
2x(x + 5) - 9 (x + 5) = 0
(x + 5)(2x - 9) = 0
Step 5:
Now, use the zero product property to find the solution.
(x + 5)(2x - 9) = 0
x + 5 = 0 or 2x - 9 = 0
x = -5 or x = 9/2
Step 6:
So, the solutions are -5 and 9/2
Practice Questions:
Solve the quadratic equations using factoring method.
a) x2 + 10x – 24 = 0
b) x2 – 9x – 36 = 0
c) x2 – 49 = 0
d) 2x2 – 5x – 12 = 0
e) 6x2 – 13x + 6 = 0
Answers:
a) x = -12 ; x = 2
b) x = -3 ; x = 12
c) x = -7 ; x = 7
d) x = - 3/2 ; x = 4
e) x = 2/3 ; x = 3/2
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