Week 7 – Precalc 11

In Week 7, we learned how to solve quadratic equations using three different methods: Factoring, the Completing the Square method, and the Quadratic Formula.

A quadratic equation is any equation that can be written in the form:

, where a, b, and c are constants and a ≠ 0.

When an equation contains multiple quadratic terms (for example, and ), it cannot be solved by isolating the variable. One strategy is to factor it using the Zero Product Law. The law states that if the product of 2 numbers is 0, either number or both numbers equal 0. Question 7a is a perfect example of this strategy:

To verify the solution, simply substitute with either 1 or 5 into the original equation and calculate the answer. For each value of , the left side of the equation should equal to the right side (0 = 0); therefore, the solution is verified.

Another method, Completing the Square on a quadratic equation, is a bit trickier. Let’s solve question 8a) by using this method:

This method needs the constant term (in this case, 2) to always be on the right side of the equation (after the equal sign). In this case, we can leave it as is.

To find a “perfect square” trinomial on the left side of the equation, we must divide the coefficient of the second term () by 2 and then square that value.

Write this value on both sides of the equation. Now, we can factor the perfect square on the left side and add up all values on the right side.

The roots are: = -2 + and = -2 –

Solving a quadratic equation using the Quadratic Formula is simpler but the equation must always be in the form of:

, where a, b, and c are constants and a ≠ 0.

Question 7a) in Unit 3.5 looks confusing at first. After re-arranging the terms, the equation will look clearer.

Then, we substitute these values into the quadratic formula.

Lastly, we took an in-depth look into the quadratic formula and learned that the expression inside the radicand of the quadratic formula is called the Discriminant. By using the Discriminant, we can determine the number of solutions (or roots) of the quadratic equation without solving the equation.

As a rule of thumb, the quadratic equation has:

Two real roots when > 0
Exactly one real root when = 0
No real roots when < 0

Question 7a) in Unit 3.6 is a good example to utilize the Discriminant and figure out whether this equation has one, two, or no real roots:

49 is greater than 0; therefore, this equation has two rational real roots. It is rational because 49 is a perfect square. If there was a number that was not a perfect square, it will have two irrational real roots (e.g. ).