Solve x2 − 7x + 5 = 0 using the Quadratic Formula.

The given quadratic equation is:

\[
x^2 - 7x + 5 = 0
\]

The general form of a quadratic equation is:

\[
ax^2 + bx + c = 0
\]

Here, \( a = 1 \), \( b = -7 \), and \( c = 5 \).

The quadratic formula is:

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

Now, substituting the values of \( a \), \( b \), and \( c \):

\[
x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(1)(5)}}{2(1)}
\]

Simplify:

\[
x = \frac{7 \pm \sqrt{49 - 20}}{2}
\]

\[
x = \frac{7 \pm \sqrt{29}}{2}
\]

Thus, the two solutions are:

\[
x_1 = \frac{7 + \sqrt{29}}{2}, \quad x_2 = \frac{7 - \sqrt{29}}{2}
\]

These are the exact solutions to the quadratic equation.

Would you like further details or any clarifications?

Here are five related questions:
1. What are the approximate decimal values of \( x_1 \) and \( x_2 \)?
2. How do you derive the discriminant from the quadratic formula?
3. Can the quadratic formula be used for equations that do not factor neatly?
4. How does the value of the discriminant affect the nature of the roots?
5. What are the possible real-world applications of solving quadratic equations?

**Tip**: Always check the discriminant (\( b^2 - 4ac \)) to determine if the roots are real, imaginary, or rational.

Solve x^2 − 7x + 5 = 0 using the Quadratic Formula.

This page provides a step-by-step solution to the quadratic equation x^2 − 7x + 5 using the quadratic formula. Learn the key algebraic concepts and apply the quadratic formula theorem. Perfect for students in Grades 9-11 to understand the exact and approximate solutions of the equation.

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