medium

2 min read

Solving by Factorisation

Solving a Quadratic Equation by Factorisation: A quadratic equation in the form of $$ax^2 + bx + c = 0$$ is a quadratic equation where a, b, and c are const...

Solving a Quadratic Equation by Factorisation:

A quadratic equation in the form of $ax^2 + bx + c = 0$ is a quadratic equation where a, b, and c are constants. Solving a quadratic equation involves finding the roots of its quadratic equation.

Factorisation Method:

The factorisation method involves breaking down the quadratic equation into two linear factors. The general form of a linear factor is $ax + b = 0$ where a and b are constants.

Step 1: Split the coefficient of x:

Divide the coefficient of x into two terms, a and b, such that $a = a_1 + a_2$ and $b = a_1 - a_2$

Step 2: Factor the left-hand side of the quadratic equation:

Factor the left-hand side of the equation using the difference of squares formula: $ax^2 + bx + c = (x - a_1)(x - a_2)$

Step 3: Equate the coefficients:

Compare the coefficients of x in the numerator and denominator to obtain the following equations:

$a_1 = a_2$

$b = -2a_1$

$c = b^2 - 4ac_1$

Step 4: Solve for x:

Using the values of a_1, a_2, and c from step 3, solve for x using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Therefore, the solutions to the quadratic equation are given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Examples:

Example 1:

Solve the quadratic equation $x^2 - 6x + 5 = 0$

Step 1: Split the coefficient of x: $a = 1, b = -6, c = 5$

Step 2: Factor the left-hand side: $x^2 - 6x + 5 = (x - 1)(x - 5)$

Step 3: Equate the coefficients: $a_1 = a_2 = 1, b = -2, c = 5$

Step 4: Solve for x: $x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(5)}}{2(1)}$

$x = \frac{2 \pm \sqrt{4 - 20}}{2}$

$x = \frac{2 \pm \sqrt{-16}}{2}$

$x = -1 \text{ and } x = 5$

Therefore, the solutions are x = -1 and x = 5.