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Rearranging and solving quadratic equations


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Algebra

Rearranging and Solving Quadratic Equations

Introduction

This topic will cover how to:

  • the definition of a quadratic equation;
  • how to solve a quadratic equation when b = 0;
  • how to solve a quadratic equation by factorising; and
  • how to solve a quadratic equation using the quadratic formula.

What is a Quadratic Equation?

A quadratic equation, or second degree equation, is an algebraic equation of the form:

ax2 + bx + c = 0,

where x is a variable and a, b and c represent known numbers such that a ≠ 0 (if a = 0 then the equation is linear). These are referred to as coefficients of the equation.

The most basic quadratic equation occurs when a = 1, b = 0 and c = 0, in which case we have:

x 2 = 0,

In this case you should be able to see that the value of the variable x must be 0, as 02 = 0. However in general, determining the value of the variable (that is, solving the equation) requires a bit of working.

This topic will cover methods of solving three different types of quadratic equations.

Solving Quadratic Equations when b = 0

The first type of quadratic equation you should be able to solve is ax 2 + bx + c = 0 when b = 0; in other words, when the equation is of the form:

You can solve equations like this by following these steps:

ax 2 + c = 0

1) Rearrange the equation so that ax2 is by itself on the left hand side of the equation, by adding or subtracting the constant c from both sides, as applicable.

For example, if you wish to solve the quadratic equation 2x2 - 72 = 0 your first steps of working would be:

2x2 - 72 = 0

2x2 - 72 + 72 = 0 + 72 (undo subtraction by adding)

2x2 = 72 (simplify)

2) Divide through both sides of the equation by a (if a ≠ 1).
For example, the next steps in solving the quadratic equation 2x2 - 72 = 0 are:

2x2/2 = 72/2 (undo multiplication by dividing)

x2 = 36 (simplify)

3) Take the square root of both sides of the equation, keeping in mind that there will be both a positive and a negative solution (except when the solution is 0); it is up to you to determine whether or not both solutions are applicable in the context of the problem.

For example, the final steps in solving the quadratic equation 2x2 - 72 = 0 are:

√x2 = √36 (take square root of both sides)

x = ±6 (simplify)

Examples: Solving Quadratic Equations
when b = 0

1) x 2 - 64 = 0

2) -x 2 + 10 = 0

3) 4x 2 + 20 = 0

x 2 - 64 + 64 = 0 + 64 (undo subtraction by adding)

x2 = 64 (simplify)

√x2 = √64 (take square root of both sides)

x = ±8 (simplify)

-x 2 + 10 - 10 = 0 - 10 (undo addition by subtracting)

-x 2 = -10 (simplify)

-x 2/-1 = -10/-1 (undo multiplication by dividing)

x 2 = 10 (simplify)

x 2 = √10 (take square root of both sides)

x = ±3.16 (simplify)

4x 2 + 20 - 20 = 0 - 20 (undo addition by subtracting)

4x2 = -20 (simplify)

4x 2/4 = -20/4 (undo multiplication by dividing)

x 2 = -5 (simplify)

x 2 = √-5 (take square root of both sides)

There are no real solutions to the equation

Activity 1: Practice Questions

Solving Quadratic Equations by Factorising

The second type of quadratic equation you should be able to solve is a quadratic equation that can be factorised using one or two sets of brackets. You can solve equations like this by following these steps:

1) Factorise the quadratic equation.
For example to solve the quadratic equation
x2 + 7x + 12 = 0, factorise it as (x + 3)(x + 4) = 0

2) Solve the equation by solving the factors, since at least one of these must be equal to zero in order for the equation to equal zero.

For example to solve the quadratic equation that can be factorised as (x + 3)(x + 4) = 0, note that either

x + 3 = 0 or x + 4 = 0. Therefore x = -3 or x = -4

Examples: Rearranging and Solving Linear Equations

1) x 2 + 7x + 6 = 0

2) x 2 - 4x - 5 = 0

3) x 2 + 3x = 0

(x + 6)(x + 1) = 0

x + 6 = 0 or x + 1 = 0

x = -6 or x = -1

(x + 1)(x - 5) = 0

x + 1 = 0 or x - 5 = 0

x = -1 or x = 5

x(x + 3) = 0

x = 0 or x + 3 = 0

x = 0 or x = -3

Activity 2: Practice Questions

Solving Quadratic Equations
using the Quadratic Formula

While factorising a quadratic equation can be a straightforward way of solving it, it is not always the case that the equation can be factorised easily- or indeed at all.

In this case, you can use the quadratic formula to solve the equation. This states that for a quadratic equation of the form y = ax2 + bx + c, the value of x is given by:

x= -b±√b 2-4ac/2a

For example, for the quadratic equation x2 + 2x - 24 = 0, the quadratic formula tells us that:

x= -2±√2 2-4(1)(-24)/2(1)

x= -2±√4+96/2

x= -2±√100/2

x= -2±10/2

x= 8/2 or (-12)/2

x= 4 or -6

x= -2±√2 2-4(1)(-24)/2(1)

x= -2±√4+96/2

x= -2±√100/2

x= -2±10/2

x= 8/2 or (-12)/2

x= 4 or -6

Examples: Solving Quadratic Equations
using the Quadratic Formula

1) x 2 + 7x + 10 = 0

2) x2 - 2x - 15 = 0

x = -7±√72-4(1)(10)/2(1)

= -7±√49-40/2

= -7±√9/2

= -7± 3/2

= -4/2 or -10/2

= -2 or-5

x = 2±√(-2)2-4(1)(-15)/2(1)

= 2±√4+60/2

= 2±√64/2

= 2±8/2

= 10/2 or -6/2

= 5 or-3

Activity 3: Practice Questions

End of Topic

Congratulations, you have completed this topic.

You should now have a better understanding of rearranging and solving quadratic equations.

 


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