Solving Quadratic Equations

Solve by Factorisation

Steps:

1. Gather together all terms to one side of the equation and make it equal to 0, in the form $ax^2+bx+c=0$.

2. Factorise the quadratic using any appropriate techniques, such as:

• Finding the highest common factor

• Using patterns like $a^2-b^2=\left(a-b\right)\left(a+b\right)$ and $a^2+2ab+b^2=\left(a+b\right)^2$

• Splitting the middle term and using grouping

• For $x^2+bx+c$, finding two numbers whose sum is $b$ and whose product is $c$

• Using the cross product method

3. Use the null factor law to split the quadratic equation into two linear equations and make each linear equation equal to 0.

4. Solve the linear equations to get the solutions to the original quadratic equation.

5. Check the solutions by substituting back into the original equation.

Example:

Solve the quadratic equation:

$3y-15y^2=0$, which has two solutions. Solve for both values of $y$.

$3y\left(-5y+1\right)=0$

$3y=0$ —> $y=0$

$-5y+1=0$ —> $5y=1$ —> $y=\frac15$

$y=0$ or $y=\frac15$

Solve by Completing the Square

Algebraic method

The process for completing the square:

1. Factor out the coefficient of the $x^2$ term, unless it is 1.

2. Halve the coefficient of $x$ and square it, add and subtract this value.

3. Factorise the terms that now form a perfect square.

4. Combine the constant and term left over from completing the square.