Calculating quadratic equation
ax2 + bx + c = 0
In algebra, a quadratic equation is any polynomial equation of the second degree with the following form:
where x is an unknown, a is referred to as the quadratic coefficient, b the linear coefficient, and c the constant. The numerals a, b, and c are coefficients of the equation, and they represent known numbers. For example, a cannot be 0, or the equation would be linear rather than quadratic. A quadratic equation can be solved in multiple ways, including factoring, using the quadratic formula, completing the square, or graphing. Only the use of the quadratic formula, as well as the basics of completing the square, will be discussed here (since the derivation of the formula involves completing the square). Below is the quadratic formula, as well as its derivation.
Example
Here's an example for solving a quadratic equation:
"Solve the quadratic equation: \( x^2 - 5x + 6 = 0 \)"
The formula to solve a quadratic equation is the quadratic formula:
\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
For the equation \( x^2 - 5x + 6 = 0 \), we have the following values:
\( a = 1 \), \( b = -5 \), and \( c = 6 \).
Plugging these values into the quadratic formula:
\( x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(6)}}{2(1)} \)
\( x = \frac{5 \pm \sqrt{25 - 24}}{2} \)
\( x = \frac{5 \pm \sqrt{1}}{2} \)
\( x = \frac{5 \pm 1}{2} \)
Now, solving for both the plus and minus values:
For the plus sign: \( x = \frac{5 + 1}{2} = \frac{6}{2} = 3 \)
For the minus sign: \( x = \frac{5 - 1}{2} = \frac{4}{2} = 2 \)
The solutions to the equation are: \( x = 3 \) and \( x = 2 \)
This demonstrates how the quadratic formula can be used to solve quadratic equations.
| Equation | Formula | Solutions | Values |
|---|---|---|---|
| \( x^2 - 5x + 6 = 0 \) | \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) | Solutions: \( x = 3 \), \( x = 2 \) | Values: \( a = 1 \), \( b = -5 \), \( c = 6 \) |
| \( 2x^2 + 3x - 2 = 0 \) | \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) | Solutions: \( x = 0.5 \), \( x = -2 \) | Values: \( a = 2 \), \( b = 3 \), \( c = -2 \) |
| \( x^2 + 4x + 4 = 0 \) | \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) | Solutions: \( x = -2 \) (repeated root) | Values: \( a = 1 \), \( b = 4 \), \( c = 4 \) |
| \( x^2 - 3x - 10 = 0 \) | \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) | Solutions: \( x = 5 \), \( x = -2 \) | Values: \( a = 1 \), \( b = -3 \), \( c = -10 \) |
| \( 3x^2 - 12x + 9 = 0 \) | \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) | Solutions: \( x = 3 \) (repeated root) | Values: \( a = 3 \), \( b = -12 \), \( c = 9 \) |
