Solving Quadratic Equations
While the ultimate goal is the same, to determine the value(s) that hold true for the equation, solving quadratic equations requires much more than simply isolating the variable, as is required in solving linear equations. This piece will outline the different types of quadratic equations, strategies for solving each type, as well as other methods of solutions such as Completing the Square and using the Quadratic Formula. Knowledge of factoring perfect square trinomials and simplifying radical expression are needed for this piece. Let’s take a look!
Standard Form of a Quadratic Equation
ax2+ bx+c=0
Where a, b, and c are integers and a≥1
I. To solve an equation in the form ax2+c=k, for some value k. This is the simplest quadratic equation to solve, because the middle term is missing.
Strategy: To isolate the square term and then take the square root of both sides.
Ex. 1) Isolate the square term, divide both sides by 2
Take the square root of both sides
2x2=40
2x22= 40 2 x2 =20
Remember there are two possible solutions x2= 20
Simplify radical; Solutions x= ± 20 x=± 25 (Please refer to previous instructional materials Simplifying Radical Expressions )
II. To solve a quadratic equation arranged in the form ax2+ bx=0.
Strategy: To factor the binomial using the greatest common factor (GCF), set the monomial factor and the binomial factor equal to zero, and solve.
Ex. 2) 12x2- 18x=0 6x2x-3= 0 Factor using the GCF 6x=0 2x-3=0 Set the monomial and binomial equal to zero x=0 x= 32 Solutions * In some cases, the GCF is simply the variable with coefficient of 1. III. To solve an equation in the form ax2+ bx+c=0, where the trinomial is a perfect square. This too is a simple quadratic equation to solve, because it factors into the form m2=0, for some binomial m.
(For factoring
Standard Form of a Quadratic Equation
ax2+ bx+c=0
Where a, b, and c are integers and a≥1
I. To solve an equation in the form ax2+c=k, for some value k. This is the simplest quadratic equation to solve, because the middle term is missing.
Strategy: To isolate the square term and then take the square root of both sides.
Ex. 1) Isolate the square term, divide both sides by 2
Take the square root of both sides
2x2=40
2x22= 40 2 x2 =20
Remember there are two possible solutions x2= 20
Simplify radical; Solutions x= ± 20 x=± 25 (Please refer to previous instructional materials Simplifying Radical Expressions )
II. To solve a quadratic equation arranged in the form ax2+ bx=0.
Strategy: To factor the binomial using the greatest common factor (GCF), set the monomial factor and the binomial factor equal to zero, and solve.
Ex. 2) 12x2- 18x=0 6x2x-3= 0 Factor using the GCF 6x=0 2x-3=0 Set the monomial and binomial equal to zero x=0 x= 32 Solutions * In some cases, the GCF is simply the variable with coefficient of 1. III. To solve an equation in the form ax2+ bx+c=0, where the trinomial is a perfect square. This too is a simple quadratic equation to solve, because it factors into the form m2=0, for some binomial m.
(For factoring