Solving Quadratic Equations by the Square Root Property

Learning Outcomes

  • Use the square root property to solve a quadratic equation

The principal square root of a nonnegative number [latex]x[/latex] is defined to be the nonnegative number [latex]a[/latex] such that [latex]a^2=x[/latex]. We write

[latex]\sqrt{x} = a[/latex] if [latex]a^{2} = x[/latex] where [latex]x \geq 0, a \geq0[/latex].

For example, [latex]\sqrt{9} = 3[/latex] since [latex]3^{2}=9[/latex].

A quadratic equation is an equation which can be written in the form

[latex]ax^2 +bx +c =0[/latex] where [latex]a,b,[/latex] and [latex]c[/latex] are constants, and [latex]a \neq 0[/latex].

Quadratic equations which can be written in the form [latex]X^2 =k[/latex], where [latex]X[/latex] is any variable expression and [latex]k[/latex] is a nonnegative number can be solved by the square root property.

The Square Root Property

If [latex]x^{2}=a[/latex], then [latex]x=\sqrt{a}[/latex] or [latex]-\sqrt{a}[/latex].

 

The solutions of [latex]x^2=a[/latex] are called the square roots of a.

  • When a is positive, [latex]a > 0[/latex], [latex]x^2=a[/latex] has two solutions, [latex]+\sqrt{a},-\sqrt{a}[/latex]. [latex]+\sqrt{a}[/latex] is the nonnegative square root of a, and [latex]-\sqrt{a}[/latex] is the negative square root of a.
  • When a is negative, [latex]a < 0[/latex], [latex]x^2=a[/latex] has no solutions.
  • When a is zero, [latex]a = 0[/latex], [latex]x^2=a[/latex] has one solution: [latex]a = 0[/latex]

Example

Solve using the square root property. [latex]x^{2}=16[/latex]

In the example above, you can take the square root of both sides easily because there is only one term on each side. In some equations, you may need to isolate the second-degree (squared) expression before applying the square root property.

In our first video, we will show more examples of using the square root property to solve a quadratic equation.

Example

Solve [latex]3x^2-1=74[/latex].

Try It

Sometimes more than just a single variable is being squared.

Example

Solve [latex](x-1)^2=16[/latex].

In the next video, you will see more examples of using square roots to solve quadratic equations.