A quadratic equation in the variable of x is an equation of the general form ax^2 + bx + c = 0 where a, b and c are coefficients, a ≠ 0. The letter "x" is the variable or unknown.
The constants a is called coefficient of x^2, b is called linear coefficient (coefficient of x) and c is called the constant term.
Similarly, 2x^2 – 3x + 1 = 0, 3x^2-5x -2 = 0 and x^2-x- 300 = 0 are also quadratic equations. In fact, any equation of the form p(x) = 0.
Where p(x) is a polynomial of degree 2, is the quadratic equation. This is the simplest way to the factorize would be to find the common factors.
Solving quadratic equation using quadratic formula:
X = [-b ± √(b^2 - 4ac) ]/ 2a
Solve the quadratic equation using factoring method:
Example problem 1:
Solve: x^2-10x + 9=0
Solution:
Factoring by splitting the middle term method,
Here a = coefficient of x^2 = 1
b = coefficient of x = -10
c = constant term = 9
We find a × c = 1× 9 = 9 = -1*-9, (-1) + (-9) = -10 = b.
x^2 - 10x + 9 = 0
x^2 + (- 1 - 9)x + 9 = 0
x^2 – 1 x – 9 x + 9 = 0
x (x - 1) – 9 (x - 1) = 0
(x – 1) (x – 9) = 0
x = 1, 9
So, the answer is x = 1, 9.
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Solve the quadratic equation using quadratic formula:
Example problem 2:
Solve: x^2-4x + 3=0
Solution:
Coefficients are: a = 1, b = -4, c = 3
Quadratic Formula: x = [-b ± √ (b^2 - 4ac)] / 2a
Put in a, b and c: x = [-(-4) ± √ [(-4)2-4×1× (3)] / (2×1)
Solve: x = [4 ± √ (16-12)]/2
x = [4 ± √ (4)]/2
x = (4 ± 2)/2
x= (4+2)/2 x = (4-2)/2
x= (6)/2 x= (2)/2
x=3 x=1
So, the answer is x=3, 1.
Practice problems on quadratic equations:
1) Solve the quadratic equation using quadratic formula: x^2 +10x + 9=0 (Answer: x = -1, -9).
2) Solve the quadratic equation using quadratic formula: x^2+ 5x + 4=0 (Answer: x = -1, -4).
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