Question

При решении уравнения х2 + 72 - 17х = 0 восьмиклассник нашел корни 5 и 7. Верно ли он решил уравнение?

Answer 1

Х2+72-17х=0
Если мы подставим 5:
5×2+72-17×5=10+72-85=82-85= -3;
Если мы подставим 7:
7×2+72-17×7=14+72-119=86-119=33

Answer 2

Solving the Equation \(x^2 - 17x + 72 = 0\)

To determine if the eighth-grader correctly solved the equation, let's first examine the given equation and then verify the roots provided by the student.

The equation given is \(x^2 - 17x + 72 = 0\).

Verifying the Roots

The student claims to have found the roots of the equation as 5 and 7. Let's verify if these roots are correct.

To verify the roots, we can use the quadratic formula, which states that for an equation of the form \(ax^2 + bx + c = 0\), the roots are given by:

\[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\]

Substituting the values from the given equation \(x^2 - 17x + 72 = 0\), we get:

\[x = \frac{{-(-17) \pm \sqrt{{(-17)^2 - 4*1*72}}}}{{2*1}}\]

Solving this equation will give us the roots of the given quadratic equation.

Calculating the Roots

Let's calculate the roots using the quadratic formula:

\[x = \frac{{17 \pm \sqrt{{289 - 288}}}}{2}\] \[x = \frac{{17 \pm 1}}{2}\]

So, the roots of the given equation are: \[x_1 = \frac{{17 + 1}}{2} = 9\] \[x_2 = \frac{{17 - 1}}{2} = 8\]

Conclusion

The roots of the given equation \(x^2 - 17x + 72 = 0\) are 9 and 8, not 5 and 7 as claimed by the student. Therefore, the student did not correctly solve the equation.

The student's solution is incorrect.

It's important to encourage the student to review the process of finding roots for quadratic equations and offer assistance in understanding the correct method for solving such equations.