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## SS2: MATHEMATICS - 2ND TERM

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Lesson 6, Topic 1
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# Finding a Quadratic Equation whose Roots are Given

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• Finding a Quadratic Equation whose Roots are Given

If α and β are the roots of a quadratic equation, then either x = α or x = β

i.e. x – α = 0 or x – β = 0

∴ (x – α)(x – β) = 0

By expansion we have

$$\scriptsize x^2 \: – \: αx \: – \: xβ\: + \: αβ = 0$$

= $$\scriptsize x^2 \: -\: \left(\: α \: + \: β \right)x \: + \: αβ = 0$$

Note that:

(i) The coefficient of x2 is 1

(ii) α + β = Sum of the roots.

Observe that it is the coefficient of x with the sign changed

(iii) αβ = Product of roots. Observe that it is the constant term

Thus we have:

$$\scriptsize x^2 \: – \: \left(\: α \: + \: β \right)x \: + \: αβ = \scriptsize x^2\: +\: \normalsize \frac{b}{a}\scriptsize x\: +\: \normalsize \frac{c}{a}$$

Comparing coefficients, we have

$$\scriptsize \:- \left(\: α \: + \: β \right) = \normalsize \frac{b}{a} \scriptsize \: and \: αβ = \normalsize \frac{c}{a}$$

$$\scriptsize \left(\: α \: + \: β \right) = \: – \normalsize \frac{b}{a} \scriptsize \: and \: αβ = \normalsize \frac{c}{a}$$

### Example 6.1.1:

Find:

(i) a quadratic equation whose roots are 5 & -4, $$\frac{2}{5}$$ and $$\scriptsize -1 \normalsize \frac{1}{2}$$

(ii) the sum and product of roots of 15y2 – 17y + 5 = 0 and 6x2 = 2x + 20

(iii) If α = $$– \frac{1}{2}$$ is a root of the quadratic  equation 8x2 – bx – 3 = 0, find the value of

(a) b

(b) the other root, β

(c) $$\left (\frac{1}{α} \: – \: \frac{1}{β} \right)^2$$

(iv) If one root of the quadratic equation 6x2 – kx + 48 = 0 is the square of the other, find the two roots and hence find the value of k.

(i) a quadratic equation whose roots are 5 & -4, $$\frac{2}{5}$$ and $$\scriptsize -1 \normalsize \frac{1}{2}$$

Solution

a. Let x = 5

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