An incomplete quadratic equation is a quadratic equation that does not have one term from the form ax²+bx+c=0 (as long as the x² term is always present). These equations are generally easier to solve than a complete quadratic equation. Depending on the missing term, we have two types of incomplete quadratic equations.
In this article, we will learn about the two types of incomplete quadratic equations in detail. We will learn how to solve these types of equations and look at some practice examples.
How to solve incomplete quadratic equations?
To solve incomplete quadratic equations, we have to start by determining the missing term of the given quadratic equation in the form $latex ax^2+bx+c=0$. Depending on this, we can use two different methods to find the solutions to the equation.
Solving quadratic equations that do not have the term bx
These quadratic equations have the form $latex ax^2+c=0$ and do not have a bx term. To solve these equations, we need to isolate x² and then take the square root of both sides of the equation.
For example, let's solve the equation $latex x^2-16=0$. Therefore, we have to rearrange it as follows:
$latex x^2=16$
Now, we take the square root of both sides:
$latex x^2=\sqrt{16}$
$latex x^2=\pm 4$
Note: The negative solution must also be considered because $latex (-4)^2=16$.
Solving quadratic equations that do not have the term c
These quadratic equations have the form $latex ax^2+bx=0$ and do not have the constant term c. To solve these equations, we need to factor out the x on the left-hand side of the equation, form an equation with each factor, and then solve.
For example, let's solve the equation $latex x^2-7x=0$. Therefore, we factor it as follows:
$latex x(x-7)=0$
Now, we form an equation with each factor and solve:
$latex x=0 ~~$ or $latex ~~x-7=0$
$latex x=0 ~~$ or $latex ~~x=7$
Note: In this type of equation, one of the solutions will always be $latex x=0$.
Incomplete quadratic equations - Examples with answers
The methods for solving both types of incomplete quadratic equations are used in the following examples. Try to solve the problems yourself before looking at the solution.
EXAMPLES 1
Find the solutions of the equation $latex x^2-4=0$.
Solution
This quadratic equation does not have the bx term. Therefore, we can solve it by isolating the quadratic term and taking the square root of both sides of the equation:
$latex x^2-4=0$
$latex x^2=4$
$latex x=\pm\sqrt{4}$
$latex x=\pm 2$
The solutions of the equation are $latex x=2$ and $latex x=-2$.
EXAMPLES 2
Find the solutions of the equation $latex x^2+4x=0$.
Solution
This equation does not have the constant term c. Therefore, to solve it, we have to factor the x and form an equation with each factor:
$latex x^2+4x=0$
$latex x(x+4)=0$
$latex x=0 ~~$ or $latex ~~x+4=0$
$latex x=0 ~~$ or $latex ~~x=-4$
The solutions of the equation are $latex x=0$ and $latex x=-4$.
EXAMPLES 3
Find the roots of the equation $latex x^2-10=0$.
Solution
To solve this equation, we have to isolate the quadratic term and take the square root of both sides:
$latex x^2-10=0$
$latex x^2=10$
$latex x=\pm\sqrt{10}$
The roots of the equation are $latex \sqrt{10}$ and $latex -\sqrt{10}$.
EXAMPLES 4
What are the roots of the equation $latex x^2-10x=0$?
Solution
To solve this equation, we have to factor the x of both terms and then form an equation with each factor of the quadratic equation:
$latex x^2-10x=0$
$latex x(x-10)=0$
$latex x=0 ~~$ or $latex ~~x-10=0$
$latex x=0 ~~$ or $latex ~~x=10$
The roots of the equation are $latex x=0$ and $latex x=10$.
EXAMPLES 5
Prove that the equation $latex x^2+9=0$ has no real roots.
Solution
We can isolate the quadratic term and then take the square root of both sides of the equation:
$latex x^2+9=0$
$latex x^2=-9$
$latex x=\pm\sqrt{-9}$
We see that we got a square root of a negative number. Therefore, the quadratic equation has no real roots.
Note: If we are using imaginary numbers, the equation has two complex roots.
EXAMPLES 6
Determine the roots of the equation $latex 4x^2+8x=0$.
Solution
In this case, we can factor the 4x term from the equation. Then, we form an equation with each term and solve:
$latex 4x^2+8x=0$
$latex 4x(x+2)=0$
$latex 4x=0 ~~$ or $latex ~~x+2=0$
$latex x=0 ~~$ or $latex ~~x=-2$
The solutions of the equation are $latex x=0$ and $latex x=-2$.
EXAMPLES 7
Find the solutions of the equation $latex 4x^2-20=0$.
Solution
To solve this equation, we need to isolate x² and then take the square root of both sides of the equation:
$latex 4x^2-20=0$
$latex 4x^2=20$
$latex x^2=5$
$latex x=\pm\sqrt{5}$
The solutions of the equation are $latex \sqrt{5}$ and $latex -\sqrt{5}$.
Incomplete quadratic equations - Practice problems
Solve the following problems by applying the methods of solving incomplete quadratic equations.
See also
Interested in learning more about quadratic equations? Take a look at these pages:
