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Solving Quadratic Equations I: Solving Quadratic Equations with Surds and Imaginary Numbers In this paper, we'll investigate how to solve quadratic expressions and equations with surds and imaginary numbers. What you will need: A basic understanding of algebra and how to multiply brackets. Knowledge of the Quadratic Formula: 2 4 2 b b ac x a
What you will learn: How to solve any and all quadratic expressions. This guide will be useful if you:
- are learning advanced quadratics
- have mastered the quadratic formula and need answers with surds and imaginary numbers Topics covered: Writing answers to quadratic equations as square roots Finding answers to quadratic equations involving imaginary numbers.
Surds Let's consider the quadratic equation x^2 -4=0. We can probably factor and solve this very quickly. The factors are ( x -2) and ( x +2), and the solutions are x =2 or x =-2, which we could write as x =2. Now let's consider the quadratic equation x^2 -3=0. We can see that the factors are x 3 and the solutions are x =3. Notice we don't write the answer as a decimal; such an answer would be approximate, not exact. Take 3 to three decimal places; it's 1.732; if we square 3 we get 3, whereas if we square 1.732 we get 2.999 which is not 3 exactly. So, to preserve accuracy, we can simply leave the answer as a square root – we call this a surd form.
Try it yourself: Solve the quadratic equation x^2 -6=0. Write your answer in surd form. Do you get the answer x = 2 3? Solve the quadratic equation x^2 -5=0. Write your answer in surd form. Do you get the answer x =5? Solve the quadratic equation x^2 -18=0. Write your answer in surd form. Do you get the answer x = 3 2?
Imaginary Numbers Let's consider the equation x^2 +1=0, and try to find the solutions. It doesn't look like a difference of squares, and we can't seem to factorise it, so we'll use the quadratic formula: 2 4 2 b b ac x a
Here, a=1, b=0, c=1, and we can substitute these values in:
2 0 0 4 1 1 2 1 x
We can simplify this: 4 2 x
We can factorise -4 as -1x2x2: 1 2 2 2 x
Since the square root of two times itself is two, we can simplify this as: 2 1 2 x
which leads to: x = ± - 1 And here we might stop, because the answer seems impossible, there is no real number that multiplies by itself to give -1; such a number needs to be positive and negative at the same time. However, there are such a thing as imaginary numbers, which are defined by i =-1. So, our answer becomes: x = ± i
Worked Example #2: Solve the equation x^2 +4=0. We'll use the quadratic formula: (^2 ) 2 b b ac x a
where a=1, b=0, c=4, leading to:
2 0 0 4 1 4 2 1 x
which leads to: 16 2 x
which leads to: 16 1 2 x
we can simplify this as: 4 2 i x
which leads us to our solution: x = ± 2 i
Try it yourself: Solve the following equations, giving your answer in surd form where necessary: x^2 +9= x^2 +3= Do you get the solutions x = 3 i and the solutions x = 3 i?
Worked Example #3: Solve the equation x^2 - x +2=0, giving your answer in surd form where necessary. We'll use the quadratic formula: (^2 ) 2 b b ac x a
where a=1, b=-1, c=2, leading to:
2 1 1 4 1 2 2 1 x
which leads to: 1 1 8 2 x
which leads to the solutions: 1 7 2 i x
Worked Example #4: Solve the equation x^2 -3 x +4=0, giving your answer in surd form where necessary. We'll use the quadratic formula: (^2 ) 2 b b ac x a
where a=1, b=-3, c=4, leading to:
2 3 3 4 1 4 2 1 x
This simplifies to: 3 9 16 2 x
which leads to: 3 7 2 x
which leads to the solutions: 3 7 2 i x
Summary: Surds are numbers with square root symbols, like these: 2, 23, 1+ 2 The imaginary number, i , is defined as (-1) Imaginary numbers are multiples of i , like these: 2 i , -3 i Complex numbers are real and imaginary numbers combined into one term: 3+2 i.
