A quadratic equation is in the form of x^2–2p… | GMAT Sprint

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PS · Problem Solving

705-805

Roots

A quadratic equation is in the form of $x^2–2px + m = 0$ , where $m$ is divisible by 5 and is less than 120. One of the roots of this equation is 7. If $p$ is a prime number and one of the roots of the equation, $x^2–2px + n = 0$ is 12, then what is the value of $p+n–m$ ?

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Explanation

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Official Explanation

Official

For the first equation

x^2–2px + m = 0

, one root is 7.

Let the other root be

r

.

By Vieta's formulas:

- Sum of roots:

7 + r = 2p

- Product of roots:

7r = m

Given

m

is divisible by 5 and less than 120,

7r

must be divisible by 5.

Possible values for

r

are 5, 10, or 15, giving

m = 35, 70, \text{or } 105

.

From

7 + r = 2p

, we have

p = \frac{7 + r}{2}

.

Since

p

is prime:

- If

r = 5

p = 6

(not prime)

- If

r = 10

p = 8.5

(not integer)…

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What this question tests

What it tests

Your handling of square and higher roots, including simplifying radicals.

Common trap

Forgetting the ± when taking a square root to solve an equation, or assuming √(a²) = a.