91Ó°ÊÓ

Factoring quadratic expressions is a method used to simplify and solve quadratic equations. A quadratic equation is typically of the form \( ax^2 + bx + c = 0 \). Our task is to transform this equation by expressing it as a product of two binomials. This makes the equation easier to solve.

To factor the given expression \( a^2 + 2a - 35 \), we need to find two numbers that multiply to the constant term \,(-35)\,, and simultaneously add up to the coefficient of the linear term \,2\,. These numbers, in this case, are \( 7 \) and \(-5 \). Hence, we rewrite the equation as \( (a + 7)(a - 5) = 0 \). This factoring simplifies the original quadratic into two linear factors which can easily be addressed in the next steps.

The next step in solving a factored quadratic equation is to find its roots. Roots are the values of the variable that make the equation true (i.e., equal to zero). With our factored expression \( (a + 7)(a - 5) = 0 \), we can now set each binomial equal to zero to find the roots.

• Set \( a + 7 = 0 \), solve for \( a \), yielding \( a = -7 \).
• Set \( a - 5 = 0 \), solve for \( a \), yielding \( a = 5 \).

By solving these simple equations, we find two values: \( a = -7 \) and \( a = 5 \). These are the roots, or solutions, to the quadratic equation \( a^2 + 2a - 35 = 0 \). These values are where the parabola represented by the equation crosses the x-axis.

Verifying solutions is an important step to ensure the accuracy of the roots obtained. To verify, substitute the roots back into the original quadratic equation and check if the equation holds true.

Let's verify the first root, \( a = -7 \):

• Substitute into the equation: \( (-7)^2 + 2(-7) - 35 \).
• Calculate: \( 49 - 14 - 35 = 0 \).
• The left side equals zero, confirming \( a = -7 \) is a valid solution.

Next, verify the second root, \( a = 5 \):

• Substitute into the equation: \( 5^2 + 2(5) - 35 \).
• Calculate: \( 25 + 10 - 35 = 0 \).
• Again, the equation is satisfied, confirming \( a = 5 \) is also a valid solution.

Reviewing each root ensures both are correctly derived from the original quadratic equation.