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WAEC Further Mathematics 2015 Paper

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Question : 18

Total: 40

If

\alpha

\beta

are the roots of

x^{2} + x - 2 = 0

, find the value of

\;\frac{1}{\alpha^{2}} + \;\frac{1}{\beta^{2}}

.

$\;\frac{5}{4}$
$\;\frac{3}{4}$
$\;\frac{1}{4}$
$\;\frac{-3}{4}$

Solution:

Concept:

Use the sum and product of roots of a quadratic equation to evaluate the required expression.

Explanation:

For the equation

x^{2} + x - 2 = 0

, we have

a = 1

,

b = 1

, and

c = -2

.

Sum of roots:

\alpha + \beta = \frac{-b}{a} = \frac{-1}{1} = -1

.

Product of roots:

\alpha\beta = \frac{c}{a} = \frac{-2}{1} = -2

.

Now write:

\frac{1}{\alpha^{2}} + \frac{1}{\beta^{2}} = \frac{\beta^{2} + \alpha^{2}}{(\alpha\beta)^{2}}

.

Find

\alpha^{2} + \beta^{2} = (\alpha + \beta)^{2} - 2\alpha\beta

.

Substitute the values:

(-1)^{2} - 2(-2) = 1 + 4 = 5

.

Therefore,

\frac{1}{\alpha^{2}} + \frac{1}{\beta^{2}} = \frac{5}{(-2)^{2}} = \frac{5}{4}

.

Answer:

\frac{5}{4}

, which is option A.

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