Question

question 13 of 23
1 point
use the discriminant to find possible solutions. do not solve.
$x^2 - 10x = 36$
select the correct response:
2 real irrational solutions
1 real rational solution
2 real rational solutions
2 non - real solutions

Explanation:

Step1: Rewrite in standard form

The quadratic equation is \(x^2 - 10x = 36\). Rewrite it in the standard form \(ax^2 + bx + c = 0\) by subtracting 36 from both sides: \(x^2 - 10x - 36 = 0\). Here, \(a = 1\), \(b = -10\), and \(c = -36\).

Step2: Calculate the discriminant

The discriminant \(D\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula \(D = b^2 - 4ac\). Substitute \(a = 1\), \(b = -10\), and \(c = -36\) into the formula:

$$
LATEXBLOCK0
$$

Step3: Analyze the discriminant

• If \(D>0\) and a perfect square, there are 2 real rational solutions.
• If \(D>0\) and not a perfect square, there are 2 real irrational solutions.
• If \(D = 0\), there is 1 real rational solution.
• If \(D<0\), there are 2 non - real solutions.

Since \(D = 244>0\) and 244 is not a perfect square (because \(\sqrt{244}\approx15.62\) and is not an integer), the equation has 2 real irrational solutions.

Answer:

2 real irrational solutions (corresponding to the option "2 real irrational solutions")