(i) In a quadratic equation in the form ax^{2} + 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 |
ax^{2} + bx + c = 0 |
Positive sign for both the factors. |
ax^{2} - bx + c = 0 |
Negative sign for both the factors. |
ax^{2} + bx - c = 0 |
Negative sign for smaller factor and positive sign for larger factor. |
ax^{2} - bx - c = 0 |
Positive sign for smaller factor and negative sign for larger factor. |
Question 1 :
Solve the following quadratic equations by factorization method.
4x^{2} - 7x - 2 = 0
Solution :
Product of coefficient of x^{2 } and constant is -8
Now, we have to decompose -8 as product of two term, such that the product of those numbers must be -8 and simplified value must be equal to -7.
Since the middle and last terms are negative, we have to put negative sign for large factor.
(4x + 1) (x - 2) = 0
4x + 1 = 0 (or) x - 2 = 0
4x = -1 (or) x = 2
x = -1/4 (or) x = 2
Hence the solutions are -1/4 and 2.
Question 2 :
Solve the following quadratic equations by factorization method.
3(p^{2} - 6) = p(p + 5)
Solution :
3p^{2} - 18 = p^{2} + 5p
3p^{2} - p^{2} - 5p - 18 = 0
2p^{2} - 5p - 18 = 0
Since the middle and last terms are negative, we have to put negative sign for large factor.
(2x + 9) (x - 2) = 0
2x + 9 = 0 (or) x - 2 = 0
2x = -9 (or) x = 2
x = -9/2
Hence the solutions are -9/2 and 2.
Question 3 :
Solve the following quadratic equations by factorization method.
√[a(a -7)] = 3√2
Solution :
√[a(a - 7)] = 3√2
Taking squares on both sides,
[√(a(a - 7))]^{2} = (3√2)^{2}
a (a - 7) = 9(2)
a^{2} - 7a = 18
a^{2} - 7a - 18 = 0
Since the middle and last terms are negative, we have to put negative sign for large factor.
(x + 1) (2x - 9) = 0
x + 1 = 0 (or) 2x - 9 = 0
x = -1 (or) 2x = 9
x = 9/2
Hence the solutions are -1 and 9/2.
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