Given the equation: x² + px - 10 = 0

Question

ScanSolve AI Tutor · Accepted Answer

Here's the solution to Question 50, verifying the steps provided:

Question 50:
Given the equation: $x^2 + px - 10 = 0$
Assume the roots are 2 and $-5$ .
Find the value of $p$ .

For a quadratic equation in the form $ax^2 + bx + c = 0$ , the following relationships hold for its roots $r_1$ and $r_2$ :

From the given equation $x^2 + px - 10 = 0$ , we have:
$a = 1$
$b = p$
$c = -10$

The assumed roots are $r_1 = 2$ and $r_2 = -5$ .

Check the product of the roots.
The product of the given roots is:
$r_1 \times r_2 = 2 \times (-5) = -10$
Using the formula for the product of roots from the equation:
$\frac{c}{a} = \frac{-10}{1} = -10$
Since the calculated product matches the formula, the assumed roots are consistent with the constant term of the equation. (Note: The student's work incorrectly showed $2 + (-5)$ for the product, but the result $-10$ was correct for the product).
Calculate the sum of the roots.
The sum of the given roots is:
$r_1 + r_2 = 2 + (-5) = -3$
Use the sum of roots formula to find $p$ .
We know that the sum of roots is also equal to $-\frac{b}{a}$ .
$r_1 + r_2 = -\frac{b}{a}$
Substitute the calculated sum and the values of $a$ and $b$ from the equation:
$-3 = -\frac{p}{1}$ $-3 = -p$
Multiply both sides by $-1$ :
$p = 3$
Therefore, the value of $p$ is 3.

Given the equation: x² + px − 10 = 0