10.1 Solve quadratic equations using the square root property

Quadratic equations are equations of the form $a{x}^2+bx+c=0$ , where $a\ne 0$ . They differ from linear equations by including a term with the variable raised to the second power. We use different methods to solve quadratic equation    s than linear equations, because just adding, subtracting, multiplying, and dividing terms will not isolate the variable.

We have seen that some quadratic equations can be solved by factoring. In this chapter, we will use three other methods to solve quadratic equations.

Solve quadratic equations of the form ax ² = k Using the square root property

We have already solved some quadratic equations by factoring. Let’s review how we used factoring to solve the quadratic equation $x^2=9$ .

We can easily use factoring to find the solutions of similar equations, like $x^2=16$ and $x^2=25$ , because 16 and 25 are perfect squares. But what happens when we have an equation like $x^2=7$ ? Since 7 is not a perfect square, we cannot solve the equation by factoring.

These equations are all of the form $x^2=k$ .
We defined the square root of a number in this way:

This leads to the Square Root Property    .

Notice that the Square Root Property    gives two solutions to an equation of the form $x^2=k$ : the principal square root of $k$ and its opposite. We could also write the solution as $x=\pm \sqrt{k}$ .

Now, we will solve the equation $x^2=9$ again, this time using the Square Root Property.

What happens when the constant is not a perfect square? Let’s use the Square Root Property to solve the equation $x^2=7$ .

How to solve a quadratic equation of the form ${ax}^2=k$ Using the square root property

Solve: $x^2-48=0$ .

Solution

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To use the Square Root Property, the coefficient of the variable term must equal 1. In the next example, we must divide both sides of the equation by 5 before using the Square Root Property.