Generally we have two types of quadratic equation.
The general form of a quadratic equation is
ax2 + bx + c = 0
In a quadratic equation, leading coefficient is nothing but the coefficient of x2.
(i) In a quadratic equation in the form ax2 + bx + c = 0, if the leading coefficient is 1, we have to decompose the constant term "c" into two factors.
(ii) The product of the two factors must be equal to the constant term "c" and the addition of two factors must be equal to the coefficient of x, that is "b".
(iii) If p and q are the two factors of the constant term c, then we have to factor the quadratic equation using p and q as shown below.
(x + p)(x + q) = 0
(iv) Solving the above equation, we get
x = -p and x = -q
Quadratic Equation |
Signs of Factors |
ax2 + bx + c = 0 |
Positive sign for both the factors. |
ax2 - bx + c = 0 |
Negative sign for both the factors. |
ax2 + bx - c = 0 |
Negative sign for smaller factor and positive sign for larger factor. |
ax2 - bx - c = 0 |
Positive sign for smaller factor and negative sign for larger factor. |
Solve the following quadratic equation by factoring :
x2 + 17 x + 60 = 0
Solution :
The given quadratic equation is in the form of
ax2 + bx + c = 0
Check whether the coefficient of x2 is 1 or not.
Because the coefficient of x2 is 1, we have to decompose 60 into two factors as shown below.
Because the constant term 60 is having positive sign, both the factors must be positive.
In the above four pairs of factors, we have to select the a pair of factors such that the product of two factors is equal to the constant term "+60" and the addition of two factors is equal to the coefficient of x, that is "+17".
Now, factor the given quadratic equation and solve for x as shown below.
(x + 12)(x + 5) = 0
x + 12 = 0 or x + 5 = 0
x = -12 or x = -5
So, the solution is {-12, -5}.
(i) In a quadratic equation in the form ax2 + bx + c = 0, if the leading coefficient is not 1, we have to multiply the coefficient of x2 and the constant term. That is "ac". Then, decompose "ac" into two factors.
(ii) The product of the two factors must be equal to "ac" and the addition of two factors must be equal to the coefficient of x, that is "b".
(iii) Divide the two factors by the coefficient of x2 and simplify as much as possible.
(iv) Write the remaining number along with x (This is explained in the following example).
Solve the following quadratic equation by factoring :
2x2 + x - 6 = 0
Solution :
The given quadratic equation is in the form of
ax2 + bx + c = 0
Here, the coefficient of x2 is 1 or not.
Multiply the coefficient of x2 and the constant term "-6".
That is,
2 ⋅ (-6) = -12
Decompose -12 into two factors such that the product of two factors is equal to -12 and the addition of two factors is equal to the coefficient of x, that is 1.
Then, the two factors of -12 are
4 and -3
Now we have to divide the two factors 4 and -3 by the coefficient of x2, that is 2.
Now, factor the given quadratic equation and solve for x as shown below.
(x + 2)(2x - 3) = 0
x + 2 = 0 or 2x - 3 = 0
x = -2 or x = 3/2
x = -2 or x = 1.5
So, the solution is {-2, 1.5}.
Problem 1 :
Solve the quadratic equation by factoring :
x2 – 5x – 24 = 0
Solution :
In the given quadratic equation, the coefficient of x2 is 1.
Decompose the constant term -24 into two factors such that the product of the two factors is equal to -24 and the addition of two factors is equal to the coefficient of x, that is 5.
Then, the two factors of -24 are
+3 and -8
Factor the given quadratic equation using +3 and -8 and solve for x.
(x + 3)(x - 8) = 0
x + 3 = 0 or x - 8 = 0
x = -3 or x = 8
So, the solution is {-3, 8}.
Problem 2 :
Solve the quadratic equation by factoring :
3x2 – 5x – 12 = 0
Solution :
In the given quadratic equation, the coefficient of x2 is not 1.
So, multiply the coefficient of x2 and the constant term "-12".
3 ⋅ (-12) = -36
Decompose -36 into two factors such that the product of two factors is equal to -36 and the addition of two factors is equal to the coefficient of x, that is -5.
Then, the two factors of -36 are
+4 and -9
Now we have to divide the two factors 4 and -9 by the coefficient of x2, that is 3.
Now, factor the given quadratic equation and solve for x as shown below.
(3x + 4)(x - 3) = 0
3x + 4 = 0 or x - 3 = 0
x = -4/3 or x = 3
So, the solution is {-4/3, 3}.
Problem 3 :
Solve the quadratic equation by factoring :
(x + 3)2 - 81 = 0
Solution :
(x + 3)2 - 81 = 0
Subtract 81 from each side.
(x + 3)2 - 81 = 0
(x + 3)2 - 92 = 0
Using the algebraic identity a2 - b2 = (a + b)(a - b), factor the polynomial on the right side.
[(x + 3) + 9][(x + 3) - 9] = 0
[x + 3 + 9][x + 3 - 9] = 0
(x + 12)(x - 6) = 0
x + 12 = 0 or x - 6 = 0
x = -12 or x = 6
So, the solution is {-12, 6}.
Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.
If you have any feedback about our math content, please mail us :
v4formath@gmail.com
We always appreciate your feedback.
You can also visit the following web pages on different stuff in math.
WORD PROBLEMS
Word problems on simple equations
Word problems on linear equations
Word problems on quadratic equations
Area and perimeter word problems
Word problems on direct variation and inverse variation
Word problems on comparing rates
Converting customary units word problems
Converting metric units word problems
Word problems on simple interest
Word problems on compound interest
Word problems on types of angles
Complementary and supplementary angles word problems
Trigonometry word problems
Markup and markdown word problems
Word problems on mixed fractrions
One step equation word problems
Linear inequalities word problems
Ratio and proportion word problems
Word problems on sets and venn diagrams
Pythagorean theorem word problems
Percent of a number word problems
Word problems on constant speed
Word problems on average speed
Word problems on sum of the angles of a triangle is 180 degree
OTHER TOPICS
Time, speed and distance shortcuts
Ratio and proportion shortcuts
Domain and range of rational functions
Domain and range of rational functions with holes
Graphing rational functions with holes
Converting repeating decimals in to fractions
Decimal representation of rational numbers
Finding square root using long division
L.C.M method to solve time and work problems
Translating the word problems in to algebraic expressions
Remainder when 2 power 256 is divided by 17
Remainder when 17 power 23 is divided by 16
Sum of all three digit numbers divisible by 6
Sum of all three digit numbers divisible by 7
Sum of all three digit numbers divisible by 8
Sum of all three digit numbers formed using 1, 3, 4
Sum of all three four digit numbers formed with non zero digits