There are many cases where the resolution of a quadratic equation by factoring or completing the square is not easy because it requires a lot of handling. In these cases, quadratic formula is quite useful and time saving way to solve the quadratic equations.
As we know a quadratic equation is an equation of the form ax2 + bx + c = 0 where a, b, and c are constants and a ≠ 0
For any such quadratic equation, the quadratic formula is:
| Roots(x1, x2) = | −b ± √ b2 − 4ac |
| 2a |
Step I: Arrange the Quadratic Equation in the standard form of ax2 + bx + c = 0.
Step II: Compare the quadratic equation which is to be solved with the standard form of quadratic equation and find out the values of the coefficients a, b, and c.
Step III: Put these values of a, b, and c in Quadratic formula.
| Roots(x1, x2) = | −b ± √ b2 − 4ac |
| 2a |
As you can notice in the formula. This implies, Quadratic formula calculates two values of x: x1 and x2, where
| x1 = | −b + √ b2 − 4ac |
| 2a |
| x2 = | −b - √ b2 − 4ac |
| 2a |
These two values of x for which ax2 + bx + c = 0 holds true, are called solutions of Quadratic Equation, or roots of Quadratic Equation.
Example: Solve, x2 - 10x = -24
Step I: - Convert above equation in standard form of quadratic equation, ax2 + bx + c = 0
x2 -12x = -32 is converted into x2 - 12x + 32 = 0
Step II: - Now find the value of coefficients a, b and c by comparing it with standard form of quadratic equation ax2 + bx + c = 0
Comparing x2 - 12x + 32 = 0 with ax2 + bx + c = 0, we get
a = 1,
b = -12,
c = 32
Step III: Putting values of a, b and c in the quadratic formula
| Roots(x1, x2) = | −b ± √ b2 − 4ac |
| 2a |
| x1 = | −b + √ b2 − 4ac |
| 2a |
| = | −(-12) + √ (-12)2 − 4(1)(32) | = 8 |
| 2(1) |
| x2 = | −b - √ b2 − 4ac |
| 2a |
| = | −(-12) - √ (-12)2 − 4(1)(32) | = 4 |
| 2(1) |
8 and 4 are the roots of the quadratic equation x2 -12x = -32
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