Activity
Solving an equation of the form is a similar process to the equations we have previously solved.
Notice that the quadratic term, , in the original form is replaced with .
Once the binomial is isolated, the square root property can be used. You can reason the solution by taking the square root and then appropriately isolating the variable using the two different square roots.
Solve the following quadratic equations. Explain or show your reasoning.
Enter the solution to the equation and your reasoning.
Compare your answer:
and
Because 12 and −12 are the two numbers we can square to make 144 and is the number being squared, we need to be 1 less than 12 or −12. So and .
Enter the solution to the equation and your reasoning.
Compare your answer:
and
Since 8 and −8 are the two numbers we can square to make 64 and is the number being squared, we need to be 6 more than 8 or −8. So and .
Enter the solution to the equation and your reasoning.
Compare your answer:
and
To isolate the binomial, add 30 to both sides of the equation. Since 10 and –10 are the two numbers we can square to make 100 and is the number being squared, we need to be 5 more than 10 or –10. So and .
Enter the solution to the equation and your reasoning.
Compare your answer:
and
To begin isolating the binomial, add 5 to both sides of the equation. Then divide both sides of the equation by 4. Since 3 and –3 are the two numbers we can square to make 9 and is the number being squared, we need to be 6 less than 3 or –3. So and .
Are you ready for more?
Extending Your Thinking
Solve the following quadratic equation. Be prepared to show your reasoning.
Enter the solution to the equation and your reasoning.
Compare your answer:
Two solutions: and .
To isolate the binomial, subtract 11 from both sides of the equation. This leaves . When you take the square root of both sides, you end up with . This leaves two equations to solve:
By solving each of them, you find the solutions and .
Video: Solving More Complex Quadratic Equations
Watch the following video to learn more about how to solve a quadratic equation of the form .
Self Check
Additional Resources
Solving Quadratic Equations of the Form
We can use the square root property to solve an equation of the form as well.
Notice that the quadratic term, , in the original form is replaced with .
Like before, the first step is to isolate the term that has the variable squared. In this case, a binomial is being squared.
Once the binomial is isolated, by dividing each side by the coefficient of , the square root property can be used on .
Solve .
Step 1 - Isolate the quadratic term and make its coefficient 1.
So, divide both sides of the equation by 4.
Step 2 - Use the square root property on the binomial.
Step 3 - Simplify the radical by pulling out the factors that are perfect squares.
Solve for . Rewrite the two solutions.
,
Step 4 Check the solutions.
Try it
Try It: Solving Quadratic Equations of the Form
Solve the following quadratic equations.
1.
2.
Here is how to solve these quadratic equations of the form :
- and
- and