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The equation $2x^2 - 8x + (4 - p) = 0$ has two real and distinct roots - Scottish Highers Maths - Question 2 - 2022

Question 2

The equation $2x^2 - 8x + (4 - p) = 0$ has two real and distinct roots. Determine the range of values for $p$.

Worked Solution & Example Answer:The equation $2x^2 - 8x + (4 - p) = 0$ has two real and distinct roots - Scottish Highers Maths - Question 2 - 2022

Step 1

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Answer

For a quadratic equation of the form $ax^2 + bx + c = 0$ , the discriminant $D$ is given by the formula:

$D = b^2 - 4ac$

In our case, we have:

Thus, the discriminant becomes:

$D = (-8)^2 - 4(2)(4 - p)$

Step 2

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Answer

We want the discriminant to be greater than zero for the equation to have two distinct real roots:

$D > 0$

Substituting $D$ into the inequality:

$64 - 8(4 - p) > 0$

Simplifying this: $64 - 32 + 8p > 0$ $32 + 8p > 0$ $8p > -32$ $p > -4$

Step 3

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Answer

The condition for the quadratic equation to have two real and distinct roots is:

$p > -4$

Thus, the range of values for $p$ is:

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