Solving Quadratic Equations: The Methods

Quadratic equations are equations in the form:

… where a, b, and c are constants. The solutions of quadratic equations are the values of x that make the equation true.

In this guide, we will show you how to solve quadratic equations using a number of methods...

1. Factorising

2. Difference of Two Squares

3. Quadratics in the form ax^2 + bx = 0

4. Factorising when a is greater than 1

5. Completing the square

6. Using the Quadratic Equation

7. The Discriminant

1. Factorising

When a quadratic expression is in the form x^2 + bx + c find two numbers that add to give b and multiply to give c.

Example One:

Solve the following quadratic equation by factorising.

This equation is already in standard form.

If the quadratic equation is factorable, you can factorise it into two binomials in the form (x + r)(x + s) = 0, where r and s are constants.

We can factorise this equation by finding two numbers that add up to –1 and multiply to get –12.

A good systematic way is to look at factors of 12:

1 & 12

2 & 6

3 & 4

The only pair that will work is 3 & 4:

3 – 4 = –1

If each bracket is set to equal 0, we can get solutions for x:

x – 4 = 0, x = 4

x + 3 = 0, x = –3

x = 4 and x = –3 are our solutions

Example Two:

Solve the following quadratic equation by factorising..

Looking at factors of 8:

1 & 8

2 & 4 (This looks good!)

The solutions are: x = 2 and x = –4

2. Difference of Two Squares

Expressions in the form

can be factorised to give:

Examples

3. Quadratics in the form ax^2 + bx = 0

Pretty easy, uh...

4. Factorising when a is greater than 1

When a quadratic is in the form

Multiply a by c = ac

Find two numbers that will add to give b and multiply to give ac.

Re-write the quadratic and replace bx with the two numbers.

Factorise in pairs.

Exercise

Factorise the following quadratic

5. Completing the square

Divide all terms by a if a > 1

Move the number term to the other side of the equation.

Complete the square on the left side of the equation - balance the c-squared value in the bracket by adding the same value to the right side of the equation.

Take the square root on both sides of the equation.

Move the number that remains on the left side of the equation to the right to find the values for x.

You could solve without changing the fractions into decimals

Key moment here;

Dividing the 0.8 by 2 as you complete the square then adding the square of the bracketed number to the other side of the equation.

Here's another example...

One more for luck...

6. Using the Quadratic Equation

The method of last resort....

If the quadratic equation is not factorable, you can use the quadratic formula - it works every time:

where a, b, and c are the constants from your quadratic equation. This formula gives you two solutions for x.

Once you have found your solutions for x, simplify them if possible. If your solutions are irrational numbers, you can leave them in that form or approximate them to a decimal.

Example:

We used this expression earlier

Using the quadratic formula...

Putting the values in:

7. The Discriminant

The discriminant of a quadratic formula is a term that appears in the quadratic formula and is given by:

The discriminant tells us information about the nature and number of solutions (or roots) of the quadratic equation.

If the discriminant is...

positive = two roots, The equation intersects the x-axis at two points.

zero = one real root & equation touches the x-axis at exactly one point.

negative = two complex conjugate roots, equation does not intersect the x-axis.