Solve Quadratic Equations by Factoring Show Learning Objective(s) · Solve equations in factored form by using the Principle of Zero Products. · Solve quadratic equations by factoring and then using the Principle of Zero Products. · Solve application problems involving quadratic equations. Introduction When a polynomial is set equal to a value (whether an integer or another polynomial), the result is an equation. An equation that can be written in the form ax2 + bx + c = 0 is called a quadratic equation. You can solve a quadratic equation using the rules of algebra, applying factoring techniques where necessary, and by using the Principle of Zero Products. The Principle of Zero Products The Principle of Zero Products states that if the product of two numbers is 0, then at least one of the factors is 0. (This is not really new.) Principle of Zero Products If ab = 0, then either a = 0 or b = 0, or both a and b are 0. This property may seem fairly obvious, but it has big implications for solving quadratic equations. If you have a factored polynomial that is equal to 0, you know that at least one of the factors or both factors equal 0. You can use this method to solve quadratic equations. Let’s start with one that is already factored.
You can check these solutions by substituting each one at a time into the original equation, (x + 4)(x – 3) = 0. You can also try another number to see what happens.
The two values that we found via factoring, x = −4 and x = 3, lead to true statements: 0 = 0. So, the solutions are correct. But x = 5, the value not found by factoring, creates an untrue statement—27 does not equal 0! Solve for x. (x – 5)(2x + 7) = 0 A) x = 5 or B) x = 5 or −7 C) x = 0 or D) x = 0 Solving Quadratics Let’s try solving an equation that looks a bit different: 5a2 + 15a = 0.
To check your answers, you can substitute both values directly into the original equation and see if you get a true sentence for each.
Both solutions check. You can use the Principle of Zero Products to solve quadratic equations in the form ax2 + bx + c = 0. First factor the expression, and set each factor equal to 0.
Note in the example above, if the common factor of 2 had been factored out, the resulting factor would be (−r + 3), which is the negative of (r – 3). So factoring out −2 will result in the common factor of (r – 3). If we had gotten (−r + 3) as a factor, then when setting that factor equal to zero and solving for r we would have gotten:
More work, but the same result as before, r = 3 or r = 2. Solve for h: h(2h + 5) = 0. A) h = 0 B) h = 2 or 5 C) h = 0 or D) h = 0 or Applying Quadratic Equations There are many applications for quadratic equations. When you use the Principle of Zero Products to solve a quadratic equation, you need to make sure that the equation is equal to zero. For example, 12x2 + 11x + 2 = 7 must first be changed to 12x2 + 11x + −5 = 0 by subtracting 7 from both sides.
The example below shows another quadratic equation where neither side is originally equal to zero. (Note that the factoring sequence has been shortened.)
If you factor out a constant, the constant will never equal 0. So it can essentially be ignored when solving. See the following example.
Solve for m: 2m2 + 10m = 48. A) m = −8 or 3 B) m = −3 or 8 C) m = 0 or −5 D) m = 0 or 5 Summary You can find the solutions, or roots, of quadratic equations by setting one side equal to zero, factoring the polynomial, and then applying the Zero Product Property. The Principle of Zero Products states that if ab = 0, then either a = 0 or b = 0, or both a and b are 0. Once the polynomial is factored, set each factor equal to zero and solve them separately. The answers will be the set of solutions for the original equation. Not all solutions are appropriate for some applications. In many real-world situations, negative solutions are not appropriate and must be discarded. How do you solve quadratic equations by factoring?To solve an quadratic equation using factoring :. 1 . Transform the equation using standard form in which one side is zero.. 2 . Factor the non-zero side.. 3 . Set each factor to zero (Remember: a product of factors is zero if and only if one or more of the factors is zero).. 4 . Solve each resulting equation.. How do you solve equations by factoring?The Solve by Factoring process will require four major steps:. Move all terms to one side of the equation, usually the left, using addition or subtraction.. Factor the equation completely.. Set each factor equal to zero, and solve.. List each solution from Step 3 as a solution to the original equation.. How do you solve quadratic equations by using the quadratic formula?How to solve a quadratic equation using the Quadratic Formula.. Write the quadratic equation in standard form, ax2 + bx + c = 0. Identify the values of a, b, c.. Write the Quadratic Formula. Then substitute in the values of a, b, c.. Simplify.. Check the solutions.. |