91Ó°ÊÓ

Quadratic equations are equations of the form ax^2 + bx + c = 0. Their solutions can be found using various methods including factoring, completing the square, and the quadratic formula. For the equation in the exercise, we used factoring to solve it. The factored form is (p+4)(p-7) = 0. This form is particularly useful because of the Zero Product Property: if the product of two numbers is zero, then at least one of the numbers must be zero. So, for solving quadratic equations like this, we break down the problem into smaller, more manageable parts.Each zero product factor is set to zero, and then the individual equations are solved. This makes finding the roots of the quadratic equation straightforward.

Factoring is the process of breaking down an expression into simpler components that, when multiplied together, produce the original expression. In the equation (p+4)(p-7) = 0, the expression (p+4) and (p-7) are factors of the quadratic equation.To factor a quadratic equation like ax^2 + bx + c = 0, we need to find two numbers that multiply to give ac and add to give b.When we identify these numbers, we split the middle term, factor by grouping, and simplify to get the factored form.Factoring makes it easier to apply the Zero Product Property and solve the quadratic equation.

The roots of an equation are the values of the variable that make the equation true. For the quadratic equation in the exercise, (p+4)(p-7)=0, the roots are the values of p that satisfy the equation. Using the Zero Product Property, we set each factor to zero: p+4=0 and p-7=0.Solving these equations gives us the roots: p=-4 and p=7. These roots represent the points where the graph of the quadratic equation intersects the x-axis. Identifying the roots helps in understanding the behavior and properties of quadratic equations.