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The equation $2x^2 + (3p - 2)x + p = 0$ has equal roots - Scottish Highers Maths - Question 5 - 2023

Question 5

The equation $2x^2 + (3p - 2)x + p = 0$ has equal roots. Determine the possible values of $p$.

Worked Solution & Example Answer:The equation $2x^2 + (3p - 2)x + p = 0$ has equal roots - Scottish Highers Maths - Question 5 - 2023

Step 1

Use the discriminant

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Answer

To determine the values of $p$ for which the quadratic equation has equal roots, we need to apply the condition that the discriminant ( $D$ ) must be equal to zero. The discriminant for the quadratic equation of the form $ax^2 + bx + c = 0$ is given by:

$D = b^2 - 4ac$

In our equation, we have:

$a = 2$
$b = 3p - 2$
$c = p$

Thus, we need to set the discriminant to zero:

$(3p - 2)^2 - 4(2)(p) = 0$

Step 2

Apply condition and express in standard quadratic form

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Answer

Substituting the values into the discriminant condition results in:

$(3p - 2)^2 - 8p = 0$

Expanding this gives:

$(9p^2 - 12p + 4) - 8p = 0$

This simplifies to:

$9p^2 - 20p + 4 = 0$

Step 3

Process for $p$

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Answer

Now, we can use the quadratic formula to find the possible values of $p$ . The quadratic formula states:

$p = rac{-b m{ ext{\pm}} ext{\sqrt}D}{2a}$

Here, $b = -20$ , $D = 0$ , and $a = 9$ . Thus, we have:

$p = rac{20 m{ ext{\pm}} 0}{2 imes 9} = rac{20}{18} = rac{10}{9}$

Therefore, the possible values of $p$ is:

$p = rac{10}{9}$

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